🔹 1. Introduction

Three-Dimensional Geometry (3D Geometry) extends coordinate geometry into space.

👉 In 2D → we use (x, y)
👉 In 3D → we use (x, y, z)

This chapter is the foundation for:

Vectors
3D Geometry (Class 12)
Physics applications

🔹 2. Coordinate Axes in 3D

In three-dimensional space, we have:

x-axis
y-axis
z-axis

👉 These three axes are mutually perpendicular and intersect at a point called the origin (0, 0, 0).

🔹 3. Coordinate of a Point

A point in space is represented as:

👉 P(x, y, z)

Where:

x → distance along x-axis
y → distance along y-axis
z → distance along z-axis

🔹 4. Octants

3D space is divided into 8 parts called octants.

👉 Based on signs of (x, y, z)

Example:

(+, +, +) → first octant
Other combinations → remaining octants

🔹 5. Distance Between Two Points

Distance between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):

$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2}$ d=(x2−x1)2+(y2−y1)2+(z2−z1)2

🔹 6. Section Formula in 3D

Coordinates of point dividing PQ in ratio m:n:

✔️ Internal Division

$\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right)$ (m+nmx2+nx1,m+nmy2+ny1,m+nmz2+nz1)

✔️ External Division

$\left(\frac{mx_2 – nx_1}{m-n}, \frac{my_2 – ny_1}{m-n}, \frac{mz_2 – nz_1}{m-n}\right)$ (m−nmx2−nx1,m−nmy2−ny1,m−nmz2−nz1)

🔹 7. Midpoint Formula in 3D

$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)$ (2×1+x2,2y1+y2,2z1+z2)

🔹 8. Direction Cosines (Basic Idea)

Direction cosines describe the orientation of a line in space.

✔️ Relation

$l^2 + m^2 + n^2 = 1$ l2+m2+n2=1

Where:

l, m, n are direction cosines

🔹 9. Important Results

✔️ Distance is always positive
✔️ Midpoint lies between two points
✔️ Coordinates determine position uniquely
✔️ 3D formulas are extensions of 2D

🔹 10. Applications

✔️ Physics (motion in space)
✔️ Engineering
✔️ Computer graphics
✔️ Navigation systems

🔹 11. JEE & CBSE Important Points

✔️ Distance formula is very important
✔️ Section formula frequently asked
✔️ Practice coordinate-based problems
✔️ Learn octant concept clearly
✔️ Accuracy in calculation is key

🔹 12. Common Mistakes

❌ Missing z-coordinate
❌ Sign errors
❌ Wrong formula substitution
❌ Confusing internal and external division

🔹 13. Practice Questions

Find distance between two points
Find midpoint of given points
Divide line in given ratio
Identify octant of a point
Verify direction cosines

🔹 14. Quick Revision Sheet

Point → (x, y, z)
Distance → √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]
Midpoint → ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)
Section formula (3D)
l² + m² + n² = 1

🔹 15. Conclusion

Introduction to 3D Geometry is a basic yet important chapter that prepares you for advanced topics in Class 12.

👉 Strong understanding ensures:

Better visualization
Faster problem solving
High exam scores

Intro to Three-Dimensional Geometry | Class 11