Three Dimensional Geometry

Master 3D coordinates, direction cosines, lines, and planes for JEE Mathematics.

Three-dimensional geometry extends 2D concepts to space, dealing with points, lines, and planes.

Overview

graph TD
    A[3D Geometry] --> B[Coordinates]
    A --> C[Lines]
    A --> D[Planes]
    B --> B1[Distance]
    B --> B2[Section Formula]
    C --> C1[Equations]
    C --> C2[Shortest Distance]

Coordinate System

Point in 3D: (x, y, z)

Distance Formula

$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$$

Section Formula

Point dividing line in ratio m:n:

$$\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right)$$

Direction Cosines and Ratios

Direction Cosines: $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$

Property: $l^2 + m^2 + n^2 = 1$

Direction Ratios: Any set (a, b, c) proportional to (l, m, n)

Conversion:

$$l = \frac{a}{\sqrt{a^2+b^2+c^2}}$$

Angle Between Lines

$$\cos\theta = l_1l_2 + m_1m_2 + n_1n_2$$ $$\cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$$

Straight Line in 3D

Equation Forms

Symmetric Form:

$$\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$$

Parametric Form:

$$x = x_1 + at, y = y_1 + bt, z = z_1 + ct$$

Vector Form:

$$\vec{r} = \vec{a} + \lambda\vec{b}$$

Coplanar Lines

Lines $\frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ and $\frac{x-x_2}{a_2} = \frac{y-y_2}{b_2} = \frac{z-z_2}{c_2}$

are coplanar if:

$$\begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = 0$$

Skew Lines

Non-parallel, non-intersecting lines.

Shortest Distance:

$$d = \frac{|(\vec{a_2}-\vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|}$$
JEE Tip
If lines intersect, shortest distance = 0. If parallel, use point-to-line distance.

Plane

General Equation

$$ax + by + cz + d = 0$$

Normal vector: $(a, b, c)$

Other Forms

Normal Form: $lx + my + nz = p$

Intercept Form: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

Point-Normal Form: $a(x-x_1) + b(y-y_1) + c(z-z_1) = 0$

Distance from Point to Plane

$$d = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2+b^2+c^2}}$$

Angle Between Planes

$$\cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$$

Angle Between Line and Plane

$$\sin\theta = \frac{al + bm + cn}{\sqrt{a^2+b^2+c^2}\sqrt{l^2+m^2+n^2}}$$

Practice Problems

  1. Find the distance between points (1, 2, 3) and (4, 5, 6).

  2. Find the equation of plane through (1, 1, 1) perpendicular to line with DR (2, 3, 4).

  3. Find shortest distance between lines: $\frac{x-1}{2} = \frac{y+1}{3} = z$ and $\frac{x+1}{5} = \frac{y-2}{1} = z-2$

Quick Check
How do you determine if two lines are skew?

Further Reading