Hey everyone! Welcome to the fascinating world of Three Dimensional Geometry! Get ready to expand your spatial reasoning beyond just flat surfaces.

### From 2D to 3D – A Leap into Space!

Remember how we used to pinpoint a location on a flat sheet of paper, a blackboard, or a computer screen? We used something called the Cartesian Coordinate System, where two perpendicular lines – the X-axis and the Y-axis – helped us define any point with an ordered pair (x, y). For example, (3, 2) meant move 3 units right, then 2 units up. Simple, right?

But now, let me ask you a question: Can you tell me the exact location of a tiny fly buzzing around in your room using just two numbers? What if it's near the ceiling, or halfway up a wall? Definitely not! Two numbers are simply not enough to describe its position in a three-dimensional space. We need more information, a third dimension!

This is exactly where Three Dimensional Geometry (3D Geometry) comes into play. It's about describing and analyzing shapes, points, lines, and planes, not just on a flat surface, but in actual space – the world around us!

### The Grand Stage: The 3D Cartesian Coordinate System

Just like we had two mutually perpendicular lines (X and Y axes) in 2D, in 3D, we need three! These are known as the Coordinate Axes.

1. X-axis: Typically shown horizontally.
2. Y-axis: Also horizontal, but perpendicular to the X-axis.
3. Z-axis: Vertical, perpendicular to both X and Y axes.



All three axes are mutually perpendicular, meaning each axis is perpendicular to the other two. They intersect at a single point called the Origin, which is denoted by O(0, 0, 0).

Visualizing it:


Analogy: Imagine the corner of your room where two walls meet the floor.


* One edge along the floor going away from you could be the positive X-axis.
* The other edge along the floor going to your left could be the positive Y-axis.
* The vertical edge where the two walls meet, going upwards, would be the positive Z-axis.


The origin is that precise corner point on the floor.

Right-Handed System (for JEE aspirants):
By convention, we mostly use what's called a 'right-handed' coordinate system. If you curl the fingers of your right hand from the positive X-axis towards the positive Y-axis, your thumb will point in the direction of the positive Z-axis. This might seem like a small detail now, but it's crucial for understanding concepts like vector cross products later on.

### The Coordinate Planes – Our Flat Surfaces in Space

When any two of these axes are taken together, they form a plane. These are called the Coordinate Planes:

1. XY-plane: Formed by the X and Y axes. Every point on this plane has its z-coordinate equal to 0. (Think of the floor of your room).
2. YZ-plane: Formed by the Y and Z axes. Every point on this plane has its x-coordinate equal to 0. (Think of one wall of your room).
3. ZX-plane: Formed by the Z and X axes. Every point on this plane has its y-coordinate equal to 0. (Think of the other wall of your room).

These three planes divide the entire space into eight regions, which we call Octants.

### Locating a Point (x, y, z) – Pinpointing in Space

Now, how do we give an "address" to our buzzing fly in this 3D space? We use an ordered triplet (x, y, z).


* The x-coordinate represents the directed distance of the point from the YZ-plane.
* The y-coordinate represents the directed distance of the point from the ZX-plane.
* The z-coordinate represents the directed distance of the point from the XY-plane.

Step-by-step visualization to reach a point P(a, b, c) from the origin:

1. Starting from the Origin O(0,0,0), move 'a' units along (or parallel to) the X-axis. If 'a' is positive, move along the positive X-axis; if negative, move along the negative X-axis. You are now at (a, 0, 0).
2. From this new position, move 'b' units parallel to the Y-axis. You are now at (a, b, 0). (This step takes you across the XY-plane).
3. Finally, from (a, b, 0), move 'c' units parallel to the Z-axis. You will arrive at the point P(a, b, c).



Analogy: Imagine starting at the corner of your room (the origin).


* You walk 'a' steps along one wall (X-axis).
* Then, from that point, you walk 'b' steps parallel to the other wall, across the floor (Y-axis direction). You are now at a point on the floor.
* Finally, you take 'c' steps straight up from that point on the floor towards the ceiling (Z-axis direction).


You have now reached the exact location (a, b, c) in your room!

### The Eight Octants – Dividing Our Space

The three coordinate planes divide the entire space into eight regions, just like the X and Y axes divide a 2D plane into four quadrants. Each region is called an octant. The sign of the coordinates (x, y, z) determines which octant a point lies in.

Let's list them out:

Octant Number	Region Name	x-coordinate	y-coordinate	z-coordinate
I	OXYZ	+	+	+
II	OX'YZ	-	+	+
III	OX'Y'Z	-	-	+
IV	OXY'Z	+	-	+
V	OXYZ'	+	+	-
VI	OX'YZ'	-	+	-
VII	OX'Y'Z'	-	-	-
VIII	OXY'Z'	+	-	-

Example 1: Identifying Octants
Which octant do the following points belong to?
a) P(2, 3, 4) $
ightarrow$ All positive, so it's in the First Octant (I).
b) Q(-1, 5, -2) $
ightarrow$ x is negative, y is positive, z is negative. This corresponds to the Sixth Octant (VI).
c) R(4, -1, 0) $
ightarrow$ This point lies on the XY-plane because z=0. It's not in an octant. Specifically, since x>0 and y<0, it lies in the region corresponding to the fourth quadrant of the XY-plane.

### Projections – Shadows in 3D

Understanding projections is crucial. Think of a projection as the "shadow" of a point cast onto a plane or an axis by a light source positioned infinitely far away and perpendicular to that plane/axis.

If we have a point P(x, y, z):

1. Projection onto Coordinate Planes:
* Projection of P(x, y, z) onto the XY-plane is P_xy(x, y, 0). (You just drop the z-coordinate).
* Projection of P(x, y, z) onto the YZ-plane is P_yz(0, y, z). (You just drop the x-coordinate).
* Projection of P(x, y, z) onto the ZX-plane is P_zx(x, 0, z). (You just drop the y-coordinate).

2. Projection onto Coordinate Axes:
* Projection of P(x, y, z) onto the X-axis is P_x(x, 0, 0).
* Projection of P(x, y, z) onto the Y-axis is P_y(0, y, 0).
* Projection of P(x, y, z) onto the Z-axis is P_z(0, 0, z).

Example 2: Projections
Find the projections of the point P(5, -3, 6) onto the coordinate planes and axes.

* Projection onto XY-plane: (5, -3, 0)
* Projection onto YZ-plane: (0, -3, 6)
* Projection onto ZX-plane: (5, 0, 6)
* Projection onto X-axis: (5, 0, 0)
* Projection onto Y-axis: (0, -3, 0)
* Projection onto Z-axis: (0, 0, 6)

### Basic Distances Related to a Point (x, y, z)

Knowing a point's coordinates allows us to easily calculate its distances from the origin, axes, and planes.

Let's consider a point P(x, y, z).

1. Distance from the Origin O(0, 0, 0):
This is a direct extension of the 2D distance formula.




2. Distance from Coordinate Planes:
The distance of a point from a plane is simply the absolute value of the coordinate perpendicular to that plane.
* Distance from XY-plane (where z=0): $|z|$
* Distance from YZ-plane (where x=0): $|x|$
* Distance from ZX-plane (where y=0): $|y|$

3. Distance from Coordinate Axes:
This is slightly different. The distance from an axis means the shortest perpendicular distance.
* Distance from X-axis: The point on the X-axis closest to P(x,y,z) is (x,0,0). So, we use the distance formula between (x,y,z) and (x,0,0).
Distance = $sqrt{(x-x)^2 + (y-0)^2 + (z-0)^2} = sqrt{0^2 + y^2 + z^2}$


Distance from X-axis = $sqrt{y^2 + z^2}$
* Similarly,
Distance from Y-axis = $sqrt{x^2 + z^2}$
Distance from Z-axis = $sqrt{x^2 + y^2}$

Example 3: Calculating Distances
Find the distance of the point A(3, -4, 5) from the origin, the coordinate planes, and the coordinate axes.

* Distance from Origin:
OA = $sqrt{3^2 + (-4)^2 + 5^2} = sqrt{9 + 16 + 25} = sqrt{50} = 5sqrt{2}$ units.

* Distance from Coordinate Planes:
* From XY-plane: $|z| = |5| = 5$ units.
* From YZ-plane: $|x| = |3| = 3$ units.
* From ZX-plane: $|y| = |-4| = 4$ units.

* Distance from Coordinate Axes:
* From X-axis: $sqrt{(-4)^2 + 5^2} = sqrt{16 + 25} = sqrt{41}$ units.
* From Y-axis: $sqrt{3^2 + 5^2} = sqrt{9 + 25} = sqrt{34}$ units.
* From Z-axis: $sqrt{3^2 + (-4)^2} = sqrt{9 + 16} = sqrt{25} = 5$ units.

CBSE vs. JEE Focus: Building Your Foundation!

For CBSE exams, a strong understanding of these fundamentals is essential for scoring well in the 3D Geometry chapter. You'll be expected to confidently:
* Identify coordinates of points.
* Locate points in different octants.
* Understand the concept of coordinate planes.
* Calculate basic distances from origin, axes, and planes.
* These basics form the bedrock for subsequent topics like lines, planes, and vectors.

For JEE Main and Advanced, these fundamentals are absolutely critical and non-negotiable. While direct questions on "which octant is this point in?" are rare, the ability to:
* Visualize a point's position in space.
* Mentally project it onto planes and axes.
* Quickly determine its distances from reference elements.
* These skills are foundational for solving complex problems involving lines, planes, spheres, shortest distances, images of points, and various vector applications. A clear mental picture is often the difference between solving a problem and getting stuck. So, practice visualizing these basic concepts until they become second nature!

Keep practicing, visualize everything, and you'll master 3D Geometry in no time!

Hello future engineers! Welcome to this deep dive into one of the foundational topics of Three-Dimensional Geometry: Coordinates of a point in space. Understanding how to locate and represent points in 3D space is absolutely crucial, not just for this unit, but for various applications in physics, engineering, and computer graphics. So, let's build this concept from the ground up, just like constructing a magnificent skyscraper!

From 1D to 3D: A Journey of Dimensions

Before we jump into the complexities of three dimensions, let's quickly refresh our memory on how we locate points in simpler dimensions:

1. 
   One Dimension (1D) - The Number Line:
   

   Imagine a straight line extending infinitely in both directions. To locate a point on this line, we need just one piece of information: its distance from a fixed point called the origin (usually denoted by O). We assign a positive sign for points on one side of the origin and a negative sign for points on the other side. For example, the point A(3) is 3 units to the right of the origin, and B(-2) is 2 units to the left.

   
   

   
   Key takeaway: One coordinate defines a point in 1D.
   

   
   


2. 
   Two Dimensions (2D) - The Cartesian Plane:
   

   Now, imagine two number lines intersecting perpendicularly at their origins. These are our familiar X-axis and Y-axis. To locate a point in this plane, we need two pieces of information: its perpendicular distance from the Y-axis (called the x-coordinate or abscissa) and its perpendicular distance from the X-axis (called the y-coordinate or ordinate). A point is represented as P(x, y). For example, P(3, 2) means 3 units along the positive X-axis and 2 units along the positive Y-axis.

   
   

   
   Key takeaway: Two coordinates define a point in 2D.
   

   
   

Following this pattern, to describe a point in three-dimensional space, intuition tells us we'll need three pieces of information, and that's precisely what we'll explore now!

The Three-Dimensional Coordinate System: Unveiling Space

To locate a point in space, we extend the concept of the Cartesian plane by adding a third axis, perpendicular to both the X and Y axes. This new axis is called the Z-axis.

1. 
   The Axes and Origin:
   
   
   
   • We have three mutually perpendicular lines: the X-axis, the Y-axis, and the Z-axis.
   
   
   • These three axes intersect at a common point, which is our origin, O(0, 0, 0).
   
   
   • The positive directions of the axes are typically chosen such that if you curl the fingers of your right hand from the positive X-axis towards the positive Y-axis, your thumb points in the direction of the positive Z-axis. This is known as the right-handed coordinate system, which is standard in mathematics and physics.
   
   
   
   

   
   Analogy: Think of a corner of your room. The floor lines meeting at the corner can be your X and Y axes, and the vertical line where the two walls meet can be your Z-axis. The corner itself is the origin.
   

   
   


2. 
   The Coordinate Planes: Dividing Space
   

   The three coordinate axes define three mutually perpendicular planes, which divide the space into eight regions:

   
   
   
   
   • 
     XY-plane: This plane contains the X-axis and the Y-axis. Every point on this plane has its Z-coordinate equal to zero (e.g., P(x, y, 0)).
     
   
   
   • 
     YZ-plane: This plane contains the Y-axis and the Z-axis. Every point on this plane has its X-coordinate equal to zero (e.g., P(0, y, z)).
     
   
   
   • 
     ZX-plane (or XZ-plane): This plane contains the Z-axis and the X-axis. Every point on this plane has its Y-coordinate equal to zero (e.g., P(x, 0, z)).
     
   
   
   
   

   
   These three planes are often referred to as the coordinate planes.
   

   
   


3. 
   Octants: The Eight Regions of Space
   

   Just as two axes divide a 2D plane into four quadrants, three coordinate planes divide 3D space into eight regions, called octants.

   
   

   The signs of the (x, y, z) coordinates determine which octant a point lies in. The first octant is where all three coordinates are positive (x > 0, y > 0, z > 0). There is no standard naming convention for the other seven octants, unlike quadrants. However, we can identify them based on the signs of their coordinates:

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   Octant Number (Informal)	x-coordinate	y-coordinate	z-coordinate	Description
   I	+	+	+	The principal octant, all positive.
   II	-	+	+	Negative X, Positive Y, Positive Z
   III	-	-	+	Negative X, Negative Y, Positive Z
   IV	+	-	+	Positive X, Negative Y, Positive Z
   V	+	+	-	Positive X, Positive Y, Negative Z
   VI	-	+	-	Negative X, Positive Y, Negative Z
   VII	-	-	-	All negative.
   VIII	+	-	-	Positive X, Negative Y, Negative Z

   
   

   
   JEE Focus: While there's no fixed numbering, being able to identify the octant by the signs of coordinates is important for quick visualization and understanding geometric properties.
   

   
   

Locating a Point P(x, y, z) in Space

Let's say we have a point P with coordinates (x, y, z). Here's how to visualize its position:

1. Start at the origin O(0, 0, 0).


2. Move |x| units along the X-axis (positive if x > 0, negative if x < 0). Let's call this point A.


3. From point A, move |y| units parallel to the Y-axis (positive if y > 0, negative if y < 0). Let's call this point B.


4. From point B, move |z| units parallel to the Z-axis (positive if z > 0, negative if z < 0). This is your point P(x, y, z).

Alternatively, you can think of it as finding a point in the XY-plane first (x, y, 0), and then moving up or down by 'z' units parallel to the Z-axis.

Definition: The coordinates (x, y, z) of a point P in space represent its perpendicular distances from the YZ-plane, ZX-plane, and XY-plane respectively.

• x = perpendicular distance from YZ-plane


• y = perpendicular distance from ZX-plane


• z = perpendicular distance from XY-plane

Projections of a Point in Space

Understanding projections is vital. When we project a point onto an axis or a plane, we're essentially finding its "shadow" on that axis or plane.

1. 
   Projection onto Coordinate Axes:
   

   The projection of a point P(x, y, z) onto an axis is the point on that axis that is closest to P.

   
   
   
   
   • Projection of P(x, y, z) onto the X-axis is (x, 0, 0).
   
   
   • Projection of P(x, y, z) onto the Y-axis is (0, y, 0).
   
   
   • Projection of P(x, y, z) onto the Z-axis is (0, 0, z).
   
   
   
   


2. 
   Projection onto Coordinate Planes:
   

   The projection of a point P(x, y, z) onto a coordinate plane is the point in that plane that is closest to P.

   
   
   
   
   • Projection of P(x, y, z) onto the XY-plane is (x, y, 0). (Set z = 0)
   
   
   • Projection of P(x, y, z) onto the YZ-plane is (0, y, z). (Set x = 0)
   
   
   • Projection of P(x, y, z) onto the ZX-plane is (x, 0, z). (Set y = 0)
   
   
   
   

Distance of a Point from Axes and Planes

Let's find the distance of a point P(x, y, z) from the coordinate axes and planes.

1. 
   Distance from Coordinate Planes:
   

   This is straightforward from the definition of coordinates:

   
   
   
   
   • Distance of P(x, y, z) from the XY-plane is |z|.
   
   
   • Distance of P(x, y, z) from the YZ-plane is |x|.
   
   
   • Distance of P(x, y, z) from the ZX-plane is |y|.
   
   
   
   

   We use absolute values because distance is always non-negative.

   
   


2. 
   Distance from Coordinate Axes:
   

   This requires a bit of thinking. Consider the distance of P(x, y, z) from the X-axis.

   
   

   The projection of P onto the X-axis is A(x, 0, 0). The distance PA is the shortest distance from P to the X-axis. This distance can be found using the distance formula in 3D (which we'll cover in detail later, but for now, remember it's a generalization of Pythagoras).

   
   

   The coordinates of P are (x, y, z) and of A are (x, 0, 0).

   
   

   Distance from X-axis = PA = √[(x-x)² + (y-0)² + (z-0)²] = √[y² + z²]

   
   

   Similarly:

   
   
   
   
   • Distance of P(x, y, z) from the Y-axis is √[x² + z²].
   
   
   • Distance of P(x, y, z) from the Z-axis is √[x² + y²].
   
   
   
   

Illustrative Examples

Let's solidify our understanding with some practical examples.

Example 1: Locate the point P(2, -3, 4) and determine its octant.

Solution:

1. Start at the origin (0,0,0).


2. Move 2 units along the positive X-axis.


3. From there, move 3 units parallel to the negative Y-axis.


4. From there, move 4 units parallel to the positive Z-axis. This is point P.

Since x = 2 (>0), y = -3 (<0), and z = 4 (>0), the point P(2, -3, 4) lies in the IV^th Octant (according to our table above: +, -, +).

Example 2: For the point Q(-1, 5, -2), find its projections onto the coordinate planes and axes.

Solution:

1. Projections onto Coordinate Planes:
   
   
   
   • Onto XY-plane: (x, y, 0) → (-1, 5, 0)
   
   
   • Onto YZ-plane: (0, y, z) → (0, 5, -2)
   
   
   • Onto ZX-plane: (x, 0, z) → (-1, 0, -2)
   
   
   
   


2. Projections onto Coordinate Axes:
   
   
   
   • Onto X-axis: (x, 0, 0) → (-1, 0, 0)
   
   
   • Onto Y-axis: (0, y, 0) → (0, 5, 0)
   
   
   • Onto Z-axis: (0, 0, z) → (0, 0, -2)
   
   
   
   

Example 3: Calculate the distance of the point R(3, -4, 5) from the coordinate planes and coordinate axes.

Solution:

Given R(x, y, z) = (3, -4, 5)

1. Distances from Coordinate Planes:
   
   
   
   • From XY-plane = |z| = |5| = 5 units.
   
   
   • From YZ-plane = |x| = |3| = 3 units.
   
   
   • From ZX-plane = |y| = |-4| = 4 units.
   
   
   
   


2. Distances from Coordinate Axes:
   
   
   
   • From X-axis = √[y² + z²] = √[(-4)² + 5²] = √[16 + 25] = √41 units.
   
   
   • From Y-axis = √[x² + z²] = √[3² + 5²] = √[9 + 25] = √34 units.
   
   
   • From Z-axis = √[x² + y²] = √[3² + (-4)²] = √[9 + 16] = √25 = 5 units.
   
   
   
   

JEE Corner: Important Considerations and Common Pitfalls

• Visualization is Key: In 3D geometry, your ability to visualize is a superpower. Always try to imagine the points, lines, and planes in space. Use real-world analogies like rooms, boxes, etc.


• Sign Convention: Pay close attention to the signs of the coordinates. They dictate the octant and thus the relative position of the point. A common mistake is to confuse which coordinate corresponds to which axis or plane.


• Projections vs. Points on Planes/Axes: A point (x,y,0) is *on* the XY-plane. It is also the *projection* of (x,y,z) onto the XY-plane. Don't confuse these terms.


• Distance vs. Coordinate: Remember that the coordinate 'x' itself is a directed distance, but 'distance from YZ-plane' is always positive, hence |x|. Similarly for axes, it's not just one coordinate.


• Origin's Coordinates: Always O(0,0,0). Points on an axis have two zero coordinates (e.g., (x,0,0) on X-axis). Points on a plane have one zero coordinate (e.g., (x,y,0) on XY-plane).


• Practice Drawing: Even simple sketches of 3D axes and points can greatly aid your understanding.

Understanding coordinates of a point in space is the bedrock of 3D geometry. Every concept, from distances and section formulae to lines, planes, and spheres, builds upon this fundamental idea. Master it, and you'll find the rest of 3D geometry much more intuitive and approachable!

Understanding the basics of Three Dimensional Geometry, especially the coordinate system, is fundamental. Remembering the order of coordinates, identification of planes, axes, and particularly the signs in different octants can sometimes be tricky. Here are some mnemonics and short-cuts to help you recall these concepts quickly and accurately during exams.

1. Order of Coordinates

The standard order for coordinates in 3D space is (x, y, z).

• 
  Short-cut: Alphabetical Order
  
  
  Just like the alphabet, 'x' comes before 'y', and 'y' comes before 'z'.
  Think of them as axes: X-axis, Y-axis, Z-axis.
  

2. Coordinate Planes

There are three main coordinate planes: XY, YZ, and ZX planes. These planes are defined by setting one coordinate to zero.

• 
  Mnemonic: "The Missing Letter is Zero"
  
  
  The name of the plane tells you which coordinate is present. The coordinate that is *not* mentioned in the plane's name is zero on that plane.
  
  
  
  • XY-plane: 'Z' is missing. So, z = 0. Any point on the XY-plane is of the form (x, y, 0).
  
  
  • YZ-plane: 'X' is missing. So, x = 0. Any point on the YZ-plane is of the form (0, y, z).
  
  
  • ZX-plane: 'Y' is missing. So, y = 0. Any point on the ZX-plane is of the form (x, 0, z).
  
  
  
  This is a common source of confusion, especially for CBSE students initially. For JEE, this must be second nature.
  

3. Points on Axes

A point lying on any axis will have zeros for the other two coordinates.

• 
  Short-cut: "Only the Axis Coordinate Survives"
  
  
  If a point lies *on* the X-axis, it only has an X-coordinate value; its Y and Z coordinates are zero. The same logic applies to the other axes.
  
  
  
  • Point on X-axis: (x, 0, 0)
  
  
  • Point on Y-axis: (0, y, 0)
  
  
  • Point on Z-axis: (0, 0, z)
  
  
  
  

4. Octants and Their Signs

The three coordinate planes divide space into eight regions called octants. Remembering the sign convention for (x, y, z) in each octant is crucial.

• 
  Systematic Recall Strategy: "2D Quadrants with a Z-Layer"
  
  
  Think of the XY-plane as your "ground". The Z-coordinate determines if you are above the ground (z > 0) or below it (z < 0).
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

  2D Quadrant (XY-Plane)	(x, y) Signs	Corresponding Octants (z > 0)	Corresponding Octants (z < 0)
  Quadrant I	(+, +)	Octant I: (+, +, +)	Octant V: (+, +, -)
  Quadrant II	(-, +)	Octant II: (-, +, +)	Octant VI: (-, +, -)
  Quadrant III	(-, -)	Octant III: (-, -, +)	Octant VII: (-, -, -)
  Quadrant IV	(+, -)	Octant IV: (+, -, +)	Octant VIII: (+, -, -)

  
  
  
  

  How to use this for quick recall:

  
  
  
  
  1. First, decide if 'z' is positive or negative. If positive, you're in Octants I-IV. If negative, you're in Octants V-VIII.
  
  
  2. Then, look at the signs of 'x' and 'y'. Match them to the 2D quadrant pattern (++, -+, --, +-).
  
  
  3. Combine 'x', 'y' signs with the 'z' sign to identify the specific octant.
  
  
  
  

Example: Where does the point P(-2, 3, -5) lie?


Here, x is negative (-), y is positive (+), and z is negative (-).


The (x,y) signs (-, +) correspond to 2D Quadrant II. Since z is negative, we combine Quadrant II with z < 0, placing the point in Octant VI.

Mastering these basic memory aids will significantly speed up your problem-solving in Three Dimensional Geometry, especially in objective type questions typical for JEE Main.

Quick Tips: Coordinates of a Point in Space

Understanding coordinates in three dimensions is fundamental to 3D Geometry. These quick tips will help you solidify your understanding and approach problems efficiently.

• Master the Basics of 3D Coordinates: A point P in space is represented by an ordered triplet (x, y, z), where x, y, and z are the perpendicular distances from the YZ, XZ, and XY planes, respectively. Remember, these coordinates uniquely define the point's position relative to the origin (0, 0, 0).


• Points on Axes and Planes:
  
  
  
  • On an Axis: A point on the X-axis will have coordinates (x, 0, 0). Similarly, for Y-axis (0, y, 0) and Z-axis (0, 0, z). Always remember that two coordinates are zero.
  
  
  • On a Coordinate Plane: A point on the XY-plane will have coordinates (x, y, 0). Similarly, for YZ-plane (0, y, z) and XZ-plane (x, 0, z). Here, one coordinate is zero. This distinction is crucial for many problems.
  
  
  
  


• Distance Formula in 3D: The distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is given by:
  
  
  $PQ = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
  
  
  It's a direct extension of the 2D formula. Don't mix up the coordinates!


• Section Formula: This formula determines the coordinates of a point that divides the line segment joining two given points in a given ratio.
  
  
  
  • Internal Division: If a point R(x, y, z) divides the line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally in the ratio m:n, then:
    
    
    $R = left( frac{m x_2 + n x_1}{m+n}, frac{m y_2 + n y_1}{m+n}, frac{m z_2 + n z_1}{m+n}
    ight)$
    
  
  
  • External Division: For external division, simply replace 'm+n' with 'm-n' and '+n' with '-n' in the numerator:
    
    
    $R = left( frac{m x_2 - n x_1}{m-n}, frac{m y_2 - n y_1}{m-n}, frac{m z_2 - n z_1}{m-n}
    ight)$
    
  
  
  
  Quick Check: For internal division, if m=n=1, it becomes the midpoint formula.


• Midpoint Formula: The midpoint of a segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is:
  
  
  $M = left( frac{x_1 + x_2}{2}, frac{y_1 + y_2}{2}, frac{z_1 + z_2}{2}
  ight)$
  
  
  This is a special case of the section formula where the ratio is 1:1.


• Centroid of a Triangle/Tetrahedron:
  
  
  
  • Triangle: For a triangle with vertices (x₁, y₁, z₁), (x₂, y₂, z₂), and (x₃, y₃, z₃), the centroid is:
    
    
    $G = left( frac{x_1 + x_2 + x_3}{3}, frac{y_1 + y_2 + y_3}{3}, frac{z_1 + z_2 + z_3}{3}
    ight)$
    
  
  
  • Tetrahedron: For a tetrahedron with vertices (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃), and (x₄, y₄, z₄), the centroid is:
    
    
    $G = left( frac{x_1 + x_2 + x_3 + x_4}{4}, frac{y_1 + y_2 + y_3 + y_4}{4}, frac{z_1 + z_2 + z_3 + z_4}{4}
    ight)$
    
  
  
  
  These are simply the average of the respective coordinates.


• JEE vs. CBSE: While CBSE focuses on direct application of these formulas, JEE problems often involve these concepts as a preliminary step in more complex problems related to planes, lines, or vectors. Be prepared to combine these basics with other topics.


• Visualisation is Key: Always try to visualize the points and their positions in 3D space. This helps in understanding the problem better and avoiding errors, especially when dealing with signs.

Mastering these fundamental formulas and conceptual understanding will build a strong base for tackling advanced topics in 3D Geometry. Practice regularly!

Intuitive Understanding: Coordinates of a Point in Space

Understanding coordinates in three-dimensional space is a fundamental concept in Geometry. It extends the familiar idea of locating points on a flat surface (2D) to pinpointing them anywhere in our real-world environment (3D).

Let's start with a quick recap of 2D coordinates:
In a 2D Cartesian system, we use two perpendicular axes, the X-axis and the Y-axis, intersecting at the origin (0,0). A point `P(x, y)` is located by moving `x` units along the X-axis and then `y` units parallel to the Y-axis. This gives us a unique position on a plane.

Stepping into the Third Dimension:
To locate a point in space, we need one more dimension, which means introducing a third axis. This is the Z-axis. The Z-axis is drawn perpendicular to both the X-axis and the Y-axis at their point of intersection, the origin.

Imagine the corner of a room:

• The line where the floor meets one wall can be your X-axis.


• The line where the floor meets the other wall (perpendicular to the first) can be your Y-axis.


• The vertical line where the two walls meet can be your Z-axis.


• The point where these three lines meet is the origin (0,0,0).

This creates a right-handed coordinate system, which is the standard used in mathematics and physics. If you curl the fingers of your right hand from the positive X-axis towards the positive Y-axis, your thumb will point in the direction of the positive Z-axis.

A point in 3D space is represented by an ordered triplet `P(x, y, z)`. Each coordinate tells us its "distance" or "position" relative to the origin along each axis:

• 
  The x-coordinate: Represents the perpendicular distance of the point from the YZ-plane (the plane formed by the Y and Z axes). It tells you how far the point is along the X-axis from the origin.
  


• 
  The y-coordinate: Represents the perpendicular distance of the point from the XZ-plane (the plane formed by the X and Z axes). It tells you how far the point is along the Y-axis from the origin.
  


• 
  The z-coordinate: Represents the perpendicular distance of the point from the XY-plane (the plane formed by the X and Y axes). It tells you how far the point is along the Z-axis from the origin.
  

Visualizing the process:
To reach a point `P(x, y, z)`:
1. Start at the origin (0,0,0).
2. Move `x` units along the X-axis.
3. From there, move `y` units parallel to the Y-axis.
4. Finally, from that position, move `z` units parallel to the Z-axis.
This sequence of movements will uniquely bring you to the point `P(x, y, z)` in space.

Just as the X and Y axes divide a 2D plane into four quadrants, the three coordinate planes (XY-plane, YZ-plane, XZ-plane) divide 3D space into eight regions called octants. The signs of `x`, `y`, and `z` determine which octant a point lies in. For instance, a point `(+, +, +)` is in the first octant.

CBSE vs. JEE Main Perspective:
For both CBSE and JEE Main, understanding the intuitive placement of a point in 3D space is foundational. While CBSE might focus more on the conceptual definition, JEE Main will quickly move to applications such as calculating the distance between two points, finding the coordinates of a point dividing a line segment, or determining properties related to planes and lines, all of which depend on a solid grasp of `(x, y, z)` coordinates.

A strong intuitive understanding of how these coordinates define a position is crucial for mastering Three-Dimensional Geometry. Keep visualizing, and you'll find it much easier!

Real World Applications of Coordinates of a Point in Space

Understanding the coordinates of a point in 3D space is not merely an abstract mathematical concept; it forms the bedrock for numerous advanced technologies and practical applications that shape our modern world. While direct questions on real-world applications are rare in JEE, appreciating these uses can significantly enhance your intuition and visualization skills, which are crucial for solving complex 3D Geometry problems.

The ability to precisely locate objects and points in three dimensions allows us to model, design, navigate, and interact with the physical world in sophisticated ways. Here are some key real-world applications:

• 
  Navigation and Global Positioning Systems (GPS):
  

  This is perhaps the most ubiquitous application. GPS systems use 3D coordinates (latitude, longitude, and altitude) to pinpoint a location on Earth. Satellites transmit signals that allow receivers (like your smartphone or car GPS) to calculate their exact position in space. This is fundamental for mapping, transportation, and tracking.

  
  


• 
  Computer Graphics and Animation:
  

  In the world of video games, movies, and virtual reality, every object, character, light source, and camera position is defined by 3D coordinates. These coordinates dictate how objects are rendered, how they move, and how they interact within a virtual 3D environment. From designing architectural models to animating complex scenes, 3D coordinates are indispensable.

  
  


• 
  Architecture and Engineering:
  

  Architects and engineers use 3D coordinates to design buildings, bridges, infrastructure, and various mechanical parts with precision. Computer-Aided Design (CAD) software extensively uses 3D coordinates to create virtual models, perform simulations, and ensure structural integrity before physical construction begins. This allows for accurate measurements, material calculations, and clash detection.

  
  


• 
  Robotics:
  

  Robots need to know their own position and the position of objects they interact with in 3D space. Whether it's a robotic arm in a manufacturing plant picking up parts or an autonomous drone navigating a complex environment, 3D coordinates are used to program movements, avoid obstacles, and perform tasks accurately.

  
  


• 
  Aerospace and Astronomy:
  

  Tracking aircraft, satellites, and spacecraft relies heavily on 3D coordinates. Air traffic control uses these to manage flight paths, while space agencies use them to monitor satellites, plan missions, and navigate probes through the solar system. Astronomers use 3D coordinates to map stars, galaxies, and other celestial bodies.

  
  


• 
  Medical Imaging:
  

  Techniques like MRI (Magnetic Resonance Imaging), CT (Computed Tomography) scans, and X-rays use 3D coordinates to generate detailed images of internal body structures. Doctors can precisely locate tumors, injuries, or anomalies within the human body, which is critical for diagnosis and surgical planning.

  
  

JEE/CBSE Relevance: While you won't typically be asked to "apply" 3D coordinates in a real-world scenario directly in exams, a strong grasp of how points are located and manipulated in 3D space is fundamental. Developing a good spatial visualization sense, aided by understanding these applications, will greatly assist you in solving problems related to lines, planes, and distances in 3D Geometry.

Keep honing your 3D visualization skills; they are truly powerful!

Common Analogies for Coordinates of a Point in Space

Understanding the concept of coordinates in three-dimensional space can be abstract initially. Analogies help bridge this gap by relating the new concept to familiar real-world scenarios. The core idea is to pinpoint an exact location using a set of measurements relative to a fixed reference point (origin) and directions (axes).

1. Transition from 1D to 2D to 3D

An excellent way to grasp 3D coordinates is by building upon your understanding of 1D and 2D systems:

• 
  1D (Number Line): Imagine a point on a straight road. You only need one number (e.g., 5 km from a landmark) to locate it. This is like the x-coordinate.
  


• 
  2D (Map): To find a house on a map, you need two pieces of information: its distance along one street (e.g., "5th house from the corner") and its distance along a perpendicular street (e.g., "on Oak Street"). This is analogous to (x, y) coordinates, defining a position on a flat plane.
  


• 
  3D (Building): Now, imagine you need to find an apartment in a building. You still need the "house number" (x) and "street" (y) on the ground, but you also need the "floor number" (z) to specify its height. This adds the third dimension, giving you (x, y, z) coordinates.
  

2. The Room Corner Analogy

This is perhaps the most common and intuitive analogy:

• 
  Origin: Pick any corner of your room where two walls and the floor meet. This corner can be considered the origin (0, 0, 0).
  


• 
  Axes: The three edges extending from this corner (along the floor and up the wall) can be considered the X, Y, and Z axes, which are mutually perpendicular.
  


• 
  Coordinates: Any point in the room (e.g., a fly in the air, a light fixture, or a specific spot on the floor) can be located by measuring its distance from the origin along these three perpendicular directions. For instance, (3 meters along one wall, 4 meters along the other wall, and 2 meters up from the floor).
  

3. GPS and Navigation

Modern GPS (Global Positioning System) offers a fantastic real-world analogy:

• 
  Latitude (x or y): Specifies how far north or south you are from the equator.
  


• 
  Longitude (y or x): Specifies how far east or west you are from the Prime Meridian.
  


• 
  Altitude/Elevation (z): Modern GPS systems also provide your height above sea level.
  

Together, these three values (latitude, longitude, and altitude) precisely pinpoint any location on or above the Earth's surface, just like (x, y, z) coordinates define a point in a Cartesian system.

4. Airplane Position

Consider an airplane in flight:

• 
  Its position projected onto the ground can be described by two coordinates (similar to latitude and longitude, or x and y on a map).
  


• 
  Its altitude (height above the ground) provides the third coordinate, the 'z' value.
  

Both ground position and altitude are essential to know the plane's exact location in the air.

These analogies help in visualizing how three independent measurements are necessary and sufficient to uniquely identify any point in three-dimensional space. This foundational understanding is crucial for tackling more complex problems in 3D Geometry for both CBSE and JEE Main.

Understanding the "Coordinates of a point in space" is fundamental to mastering Three-Dimensional Geometry. This concept serves as a building block for numerous advanced topics in both JEE Main and CBSE board examinations. Familiarity with the following related topics will significantly enhance your grasp of 3D geometry and problem-solving skills.

• 
  2D Coordinate Geometry: Before delving into 3D, a strong foundation in 2D coordinate geometry (Cartesian system, distance formula, section formula, equation of a line/circle) is essential. Many principles extend directly from 2D to 3D.
  
  
  JEE Main Relevance: While 2D itself isn't a separate topic in 3D geometry, its concepts are prerequisites. Questions often test the ability to generalize from 2D to 3D.
  


• 
  Distance Formula in 3D: This is a direct application of knowing how to represent a point in 3D space. It allows you to calculate the distance between any two points (x₁, y₁, z₁) and (x₂, y₂, z₂) using the formula $sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$.
  
  
  JEE Main/CBSE Relevance: Crucial for finding lengths of sides of figures, determining collinearity, and solving problems involving spheres.
  


• 
  Section Formula in 3D: Similar to 2D, this formula helps find the coordinates of a point that divides the line segment joining two given points in a given ratio (internally or externally). The midpoint formula is a special case of the section formula.
  
  
  JEE Main/CBSE Relevance: Frequently tested for finding centroids, incenters, or specific points on a line segment.
  


• 
  Direction Cosines and Direction Ratios: These concepts directly relate to the orientation of a line in 3D space, which in turn depends on the coordinates of points lying on or defining that line. Direction cosines are the cosines of the angles made by a line with the positive directions of the coordinate axes, while direction ratios are numbers proportional to the direction cosines.
  
  
  JEE Main/CBSE Relevance: Absolutely vital for understanding lines and planes in 3D. Almost all problems involving angles between lines/planes, or equations of lines/planes, will use these.
  


• 
  Vector Algebra: Points in 3D space can be represented as position vectors. Operations like vector addition, subtraction, dot product, and cross product are extensively used to solve problems in 3D geometry, often providing a more elegant and intuitive approach than pure Cartesian methods.
  
  
  JEE Main Relevance: There is significant overlap between Vector Algebra and 3D Geometry in JEE. Many problems can be solved using either approach, and proficiency in both is highly beneficial.
  


• 
  Equations of Lines and Planes in 3D: These are direct applications and extensions of understanding 3D coordinates, direction ratios, and direction cosines. The ability to define a line passing through given points or a plane containing specific points or having a particular normal vector relies heavily on coordinate knowledge.
  
  
  JEE Main/CBSE Relevance: These are core topics of 3D geometry, and almost every question in this unit will involve lines or planes.
  

Mastering the fundamental concept of 3D coordinates will make your journey through these interconnected topics much smoother and more successful in your exams.

Understanding coordinates of a point in space is foundational for Three Dimensional Geometry. However, several common pitfalls can lead to loss of marks in exams. Be aware of these traps and learn how to avoid them.

Common Exam Traps in Coordinates of a Point in Space

• 
  Confusing 2D and 3D Thinking:
  

  Many students instinctively revert to 2D coordinate geometry concepts when faced with 3D problems. This often leads to ignoring the z-coordinate or making assumptions based on a flat plane. For instance, when finding the distance between two points, students might mistakenly use the 2D distance formula, forgetting the component from the z-coordinates.

  
  

  Tip: Always explicitly list all three coordinates (x, y, z) for each point and ensure all formulas (distance, section, etc.) account for the third dimension. Visualise the points in 3D space to reinforce your understanding.

  
  


• 
  Sign Errors for Octants and Reflections:
  

  Identifying the correct octant or finding the reflection of a point can become tricky due to sign errors. A point (x, y, z) has its coordinates' signs determining its octant. Forgetting which coordinate changes sign during reflection across a plane or an axis is a common mistake.

  
  
  
  
  • Reflection across the xy-plane: (x, y, -z)
  
  
  • Reflection across the yz-plane: (-x, y, z)
  
  
  • Reflection across the zx-plane: (x, -y, z)
  
  
  • Reflection across the x-axis: (x, -y, -z)
  
  
  • Reflection across the y-axis: (-x, y, -z)
  
  
  • Reflection across the z-axis: (-x, -y, z)
  
  
  
  

  Tip: Remember that when reflecting across a plane, the coordinate corresponding to the axis *perpendicular* to that plane changes sign. When reflecting across an axis, the coordinates corresponding to the axes *perpendicular* to that axis change sign.

  
  


• 
  Misidentifying Planes and Axes:
  

  Students often confuse a point lying 'on an axis' with a point lying 'in a coordinate plane'.

  
  
  
  
  • A point on the x-axis has y=0 and z=0 (i.e., (x, 0, 0)).
  
  
  • A point in the xy-plane has z=0 (i.e., (x, y, 0)).
  
  
  • Similarly for other axes and planes. Forgetting which coordinate is zero can lead to incorrect problem setup.
  
  
  
  

  Tip: Clearly distinguish between points on an axis (two coordinates are zero) and points in a plane (one coordinate is zero). Always check the given condition carefully.

  
  


• 
  Incorrect Application of Section Formula:
  

  While the section formula for 3D is a direct extension of 2D, errors occur when applying it for external division, or when one of the points is the origin, or if the ratio is inverted.

  
  
  
  
  • Internal Division: $(x,y,z) = left(frac{m x_2 + n x_1}{m+n}, frac{m y_2 + n y_1}{m+n}, frac{m z_2 + n z_1}{m+n}
    ight)$
  
  
  • External Division: $(x,y,z) = left(frac{m x_2 - n x_1}{m-n}, frac{m y_2 - n y_1}{m-n}, frac{m z_2 - n z_1}{m-n}
    ight)$
  
  
  
  

  Tip: Pay close attention to the signs for internal vs. external division. Ensure the ratio (m:n) is correctly mapped to the respective points (P1 and P2).

  
  


• 
  Visualisation Gaps:
  

  3D geometry heavily relies on spatial reasoning. If you struggle to visualise the arrangement of points, axes, and planes, you might misinterpret questions or make incorrect deductions about distances, angles, or positions.

  
  

  Tip: Practice drawing 3D coordinate systems. Use your room corners as a mental model for the origin and axes. Utilise online 3D graphing tools or physical models if necessary to build strong visualisation skills.

  
  

JEE Main vs. CBSE: While CBSE emphasizes the basic understanding and application of these formulas, JEE Main often integrates these concepts into more complex problems involving lines, planes, and vectors. A strong foundation here is crucial for both, but JEE will test your ability to avoid these traps under time pressure with more layers of complexity.

By being mindful of these common traps and practicing diligently, you can significantly improve your accuracy and scores in Three Dimensional Geometry.

Key Takeaways: Coordinates of a Point in Space

Understanding the coordinates of a point in space is fundamental to Three-Dimensional Geometry. These basics serve as the building blocks for more advanced topics like lines, planes, and vectors. Mastering these concepts ensures a strong foundation for both JEE Main and board exams.

1. Representation of a Point in 3D

• A point in space is uniquely represented by an ordered triplet (x, y, z), where x, y, and z are its distances from the YZ, ZX, and XY planes, respectively.


• The intersection of the three coordinate axes (X-axis, Y-axis, Z-axis) is the origin (0, 0, 0).

2. Coordinate Planes and Axes

• Coordinate Planes:
  
  
  
  • XY-plane: z = 0 (points are of the form (x, y, 0))
  
  
  • YZ-plane: x = 0 (points are of the form (0, y, z))
  
  
  • ZX-plane: y = 0 (points are of the form (x, 0, z))
  
  
  
  


• Coordinate Axes:
  
  
  
  • X-axis: y = 0, z = 0 (points are of the form (x, 0, 0))
  
  
  • Y-axis: x = 0, z = 0 (points are of the form (0, y, 0))
  
  
  • Z-axis: x = 0, y = 0 (points are of the form (0, 0, z))
  
  
  
  


• The three coordinate planes divide the space into eight regions called octants. The sign of the coordinates (x, y, z) determines the octant in which a point lies.

3. Essential Formulas for JEE Main & CBSE

These formulas are direct extensions of their 2D counterparts and are crucial for problem-solving.

• 
  Distance Formula: The distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is given by:
  
  
  PQ = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
  


• 
  Section Formula: The coordinates of a point R that divides the line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) in the ratio m:n:
  
  
  
  • Internally: R = [(mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)]
  
  
  • Externally: R = [(mx₂ - nx₁)/(m - n), (my₂ - ny₁)/(m - n), (mz₂ - nz₁)/(m - n)]
  
  
  • Midpoint Formula (Special case of internal division where m=n=1): M = [(x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2]
  
  
  
  


• 
  Centroid of a Triangle: For a triangle with vertices A(x₁, y₁, z₁), B(x₂, y₂, z₂), and C(x₃, y₃, z₃), the centroid G is:
  
  
  G = [(x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3, (z₁ + z₂ + z₃)/3]
  


• 
  Centroid of a Tetrahedron: For a tetrahedron with vertices A(x₁, y₁, z₁), B(x₂, y₂, z₂), C(x₃, y₃, z₃), and D(x₄, y₄, z₄), the centroid G is:
  
  
  G = [(x₁ + x₂ + x₃ + x₄)/4, (y₁ + y₂ + y₃ + y₄)/4, (z₁ + z₂ + z₃ + z₄)/4]
  

4. JEE vs. CBSE Focus

Aspect	CBSE Board Exams	JEE Main
Emphasis	Understanding definitions, direct application of formulas, step-by-step solutions.	Problem-solving speed, conceptual application, integration with other topics (vectors, lines, planes).
Questions	Usually straightforward, involving a single formula application.	Can be multi-concept, requiring clever application or observation, often involves coordinate geometry in 3D.

5. Practical Tip for Success

• Visualize! Always try to visualize the points, lines, and planes in 3D space. This mental imagery greatly aids in understanding the problem and finding the correct approach, especially for complex problems.


• Practice problems involving varying scenarios: points on axes/planes, equidistant points, collinearity, and specific geometric shapes (triangles, parallelograms, etc.) in 3D.

Mastering these foundational concepts and formulas is your first step towards conquering 3D Geometry. Keep practicing!

Problem Solving Approach for Coordinates of a Point in Space

Understanding the coordinates of a point in a three-dimensional Cartesian system (X, Y, Z) is fundamental to solving problems in 3D Geometry. The problem-solving approach involves a systematic application of basic formulae and geometric reasoning.

1. Visualize the Scenario

• Whenever possible, try to visualize the points and objects in 3D space. This helps in understanding the problem and sometimes simplifies the approach.


• Sketching a rough diagram, even if it's just marking points and axes, can provide clarity.

2. Identify Given Information and Target

• Clearly list the coordinates of all given points, e.g., P($x_1, y_1, z_1$), Q($x_2, y_2, z_2$).


• Understand what the problem is asking for: Is it a distance, a ratio, a specific point (e.g., centroid, incenter), a projection, or a condition (e.g., collinearity)?

3. Select the Appropriate Formula/Concept

This is the most crucial step. Based on the target, choose the correct formula:

• Distance between Two Points: If finding the distance between P($x_1, y_1, z_1$) and Q($x_2, y_2, z_2$), use $PQ = sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$. This is a direct application for both CBSE and JEE.


• Section Formula: To find coordinates of a point R that divides the line segment PQ in ratio $m:n$:
  
  
  
  • Internally: $R = left(frac{mx_2+nx_1}{m+n}, frac{my_2+ny_1}{m+n}, frac{mz_2+nz_1}{m+n}
    ight)$
  
  
  • Externally: $R = left(frac{mx_2-nx_1}{m-n}, frac{my_2-ny_1}{m-n}, frac{mz_2-nz_1}{m-n}
    ight)$
  
  
  
  This is a standard topic for both CBSE and JEE. Often used for finding midpoints (ratio 1:1) or other division points.
  


• Centroid of a Triangle: For vertices A($x_1, y_1, z_1$), B($x_2, y_2, z_2$), C($x_3, y_3, z_3$), the centroid is $G = left(frac{x_1+x_2+x_3}{3}, frac{y_1+y_2+y_3}{3}, frac{z_1+z_2+z_3}{3}
  ight)$.


• Centroid of a Tetrahedron (JEE Specific): For vertices A($x_1, y_1, z_1$), B($x_2, y_2, z_2$), C($x_3, y_3, z_3$), D($x_4, y_4, z_4$), the centroid is $G = left(frac{x_1+x_2+x_3+x_4}{4}, frac{y_1+y_2+y_3+y_4}{4}, frac{z_1+z_2+z_3+z_4}{4}
  ight)$.


• Projection of a Point on Axes/Planes:
  
  
  
  • Projection of P($x, y, z$) on X-axis is ($x, 0, 0$).
  
  
  • Projection of P($x, y, z$) on XY-plane is ($x, y, 0$).
  
  
  • Similar for other axes and planes. This concept is vital for JEE problems involving distances from axes/planes.
  
  
  
  

4. Execute Calculations Carefully

• Substitute the given coordinates into the chosen formula.


• Perform arithmetic operations meticulously. A common mistake is sign errors or calculation blunders, especially with squares and square roots.


• For JEE Main, speed and accuracy are key. Practice mental arithmetic or use rough work efficiently.

5. Verify Your Answer

• If possible, check if your answer makes sense geometrically. For example, if you find a midpoint, check if it lies 'between' the two original points.


• Double-check the question to ensure you've answered exactly what was asked.

JEE Specific Tip: Coordinates of a point form the bedrock for advanced 3D Geometry topics like lines, planes, and spheres. Mastering these basic operations ensures you can confidently tackle complex problems that build upon them. Many JEE problems involve a combination of these basic concepts, so proficiency in each is crucial.

The CBSE board examination emphasizes a clear understanding of the fundamental concepts of three-dimensional geometry, particularly concerning the coordinates of a point in space. While JEE often delves into complex applications, CBSE focuses on direct comprehension and application of basic formulas.

CBSE Focus Areas for Coordinates of a Point in Space

For your CBSE board exams, pay close attention to the following core concepts:

• Understanding the 3D Coordinate System:
  
  
  
  • Familiarity with the three mutually perpendicular axes (X, Y, Z) and their point of intersection, the origin O(0,0,0).
  
  
  • Ability to locate a point P(x, y, z) in space based on its perpendicular distances from the coordinate planes.
  
  
  
  


• Octants:
  
  
  
  • Identification: Understanding how the three coordinate planes divide space into eight regions, called octants.
  
  
  • Sign Convention: Knowing the signs of the x, y, and z coordinates in each of the eight octants. This is a very common objective or short-answer question in CBSE.
  
  
  • Example: The first octant has (+, +, +), while the second octant (formed by negative x-axis, positive y-axis, positive z-axis) has (-, +, +).
  
  
  
  


• Coordinates on Axes and Planes:
  
  
  
  • A point on the X-axis has coordinates (x, 0, 0).
  
  
  • A point on the Y-axis has coordinates (0, y, 0).
  
  
  • A point on the Z-axis has coordinates (0, 0, z).
  
  
  • A point on the XY-plane has coordinates (x, y, 0).
  
  
  • A point on the YZ-plane has coordinates (0, y, z).
  
  
  • A point on the ZX-plane has coordinates (x, 0, z).
  
  
  
  


• Distance Formula in 3D:
  
  
  
  • Derivation and application of the formula for the distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):
    
    $PQ = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
  
  
  • This formula is crucial for proving properties of triangles (e.g., isosceles, equilateral, right-angled) and quadrilaterals, or determining collinearity of points.
  
  
  
  


• Section Formula in 3D:
  
  
  
  • Finding the coordinates of a point R that divides the line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) in the ratio m:n internally or externally.
  
  
  • Internal Division:
    
    $R = left( frac{mx_2 + nx_1}{m+n}, frac{my_2 + ny_1}{m+n}, frac{mz_2 + nz_1}{m+n}
    ight)$
  
  
  • External Division:
    
    $R = left( frac{mx_2 - nx_1}{m-n}, frac{my_2 - ny_1}{m-n}, frac{mz_2 - nz_1}{m-n}
    ight)$
  
  
  • Midpoint Formula: A special case of the section formula where m:n = 1:1.
    
    $M = left( frac{x_1 + x_2}{2}, frac{y_1 + y_2}{2}, frac{z_1 + z_2}{2}
    ight)$
  
  
  
  


• Centroid of a Triangle:
  
  
  
  • The coordinates of the centroid G of a triangle with vertices A(x₁, y₁, z₁), B(x₂, y₂, z₂), and C(x₃, y₃, z₃) are:
    
    $G = left( frac{x_1 + x_2 + x_3}{3}, frac{y_1 + y_2 + y_3}{3}, frac{z_1 + z_2 + z_3}{3}
    ight)$
  
  
  
  

CBSE vs. JEE Perspective

For CBSE, the focus is generally on direct application of these formulas and a solid conceptual grasp. You might be asked to find the distance, section point, or centroid given specific coordinates. JEE, on the other hand, often integrates these basic concepts into more complex problems involving lines, planes, and vectors, requiring a deeper analytical approach.

Example Question (CBSE Typical)

Find the coordinates of the point that divides the line segment joining the points A(2, -3, 4) and B(8, 0, 10) in the ratio 2:1 internally. Also, state the octant in which the point A lies.

Solution:

Using the internal section formula with (x₁,y₁,z₁) = (2,-3,4), (x₂,y₂,z₂) = (8,0,10), and m:n = 2:1:

x-coordinate = $(2 imes 8 + 1 imes 2) / (2+1) = (16+2)/3 = 18/3 = 6$

y-coordinate = $(2 imes 0 + 1 imes (-3)) / (2+1) = (0-3)/3 = -3/3 = -1$

z-coordinate = $(2 imes 10 + 1 imes 4) / (2+1) = (20+4)/3 = 24/3 = 8$

So, the point is (6, -1, 8).

For point A(2, -3, 4), the x-coordinate is positive, y-coordinate is negative, and z-coordinate is positive. This corresponds to the fourth octant.

Mastering these foundational concepts will ensure you perform well in the CBSE board examinations and build a strong base for more advanced topics in 3D geometry.

Understanding the coordinates of a point in space is the absolute foundation of Three-Dimensional Geometry for JEE Main. While questions rarely test this concept in isolation, a strong grasp is indispensable for tackling problems related to lines, planes, spheres, and other 3D geometric figures.

JEE Focus Areas: Coordinates of a Point in Space

The core concepts involving coordinates of a point serve as critical building blocks. JEE questions typically embed these basics into more complex problems.

• 
  Distance Formula:
  
  
  
  • For points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂), the distance is PQ = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²].
  
  
  • JEE Application: This is frequently used in locus problems (e.g., finding the equation of a surface where a point is equidistant from two given points), identifying the type of geometric figures (e.g., if three points form an equilateral triangle, if four points form a square), or finding points satisfying certain distance conditions.
  
  
  
  


• 
  Section Formula:
  
  
  
  • If a point R(x, y, z) divides the line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) in the ratio m:n, then:
    
    
    
    • Internally: x = (mx₂ + nx₁)/(m+n), y = (my₂ + ny₁)/(m+n), z = (mz₂ + nz₁)/(m+n)
    
    
    • Externally: x = (mx₂ - nx₁)/(m-n), y = (my₂ - ny₁)/(m-n), z = (mz₂ - nz₁)/(m-n)
    
    
    
    
  
  
  • Mid-point: x = (x₁ + x₂)/2, y = (y₁ + y₂)/2, z = (z₁ + z₂)/2 (when m=n=1).
  
  
  • JEE Application: Essential for finding vertices of polygons (e.g., centroid, incenter), determining collinearity of points, or finding points that divide a segment in a given ratio, often linked with vector algebra.
  
  
  
  


• 
  Centroid of a Triangle and Tetrahedron:
  
  
  
  • Triangle (A(x₁, y₁, z₁), B(x₂, y₂, z₂), C(x₃, y₃, z₃)): G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3, (z₁ + z₂ + z₃)/3)
  
  
  • Tetrahedron (A, B, C, D): G = ((x₁ + x₂ + x₃ + x₄)/4, (y₁ + y₂ + y₃ + y₄)/4, (z₁ + z₂ + z₃ + z₄)/4)
  
  
  • JEE Application: Frequently used in problems involving properties of triangles/tetrahedrons, often combined with vector concepts or finding coordinates under specific geometric conditions.
  
  
  
  


• 
  Projections and Reflections of a Point:
  
  
  
  • Projection on Coordinate Axes:
    
    
    
    • On X-axis: (x, 0, 0)
    
    
    • On Y-axis: (0, y, 0)
    
    
    • On Z-axis: (0, 0, z)
    
    
    
    
  
  
  • Projection on Coordinate Planes:
    
    
    
    • On XY-plane: (x, y, 0)
    
    
    • On YZ-plane: (0, y, z)
    
    
    • On ZX-plane: (x, 0, z)
    
    
    
    
  
  
  • Reflection across Coordinate Planes:
    
    
    
    • Across XY-plane: (x, y, -z)
    
    
    • Across YZ-plane: (-x, y, z)
    
    
    • Across ZX-plane: (x, -y, z)
    
    
    
    
  
  
  • JEE Application: Essential for visualizing and understanding 3D geometry. Questions can involve finding the image of a point or deriving properties based on projections.
  
  
  
  

CBSE vs. JEE Main:

While CBSE board exams might test direct application of these formulas, JEE Main often expects you to combine them with other concepts (e.g., vectors, equation of lines/planes, calculus for minima/maxima problems involving distances) or apply them in more abstract problem-solving scenarios. Be prepared to use these basic tools as part of a larger, multi-step solution.

Key Takeaway: Master these basic coordinate formulas thoroughly. They are the fundamental language of 3D geometry and will be implicitly used in almost every problem in this unit. Practice problems that combine these concepts to strengthen your problem-solving approach.