CBSE · Class 6-12

Class 12 · Mathematics (041)

Unit 11 · Class 12Mathematics (041)

Vectors and Three-Dimensional Geometry

Three-Dimensional Geometry

Direction cosines, lines, planes, angle between two lines/planes, shortest distance.

Lessons

4 concepts

1
Direction Cosines & Ratios
For a line with direction $(a, b, c)$ : DCs $l, m, n$ with $l^2 + m^2 + n^2 = 1$ .
2
Equation of a Line
Vector: $\vec r = \vec a + \lambda \vec b$ . Cartesian: $\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}$ .
3
Equation of a Plane
Normal form $\vec r \cdot \hat n = d$ ; Cartesian $ax + by + cz + d = 0$ .
4
Angle & Distance
Angle between two planes $=$ angle between normals. Distance of point $(x_0, y_0, z_0)$ from $ax + by + cz + d = 0$ : $\dfrac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2 + b^2 + c^2}}$ .

Skills to practise

Skill · medium

Three-Dimensional Geometry — Practice (coming soon)

Practice bank for this unit is in progress. You can still read the lessons above.