Three-Dimensional Geometry: Lines — Part 1: Direction, angles and equations: visual lesson, ISC Class 12 Mathematics

Lesson

Three-Dimensional Geometry: Lines, Part 1

Straight lines in space, from a direction to an equation.

In this lesson you will

Find the direction cosines and direction ratios of a line through two points
Write the equations of the coordinate axes and coordinate planes
Find the angle between two lines and test whether they are perpendicular or parallel
Write the vector and Cartesian equations of a line and convert between them

Part 1 at a glance.

Concept

Four skills for Part 1.
You will learn to	Topic
find direction cosines and ratios of a line through two points	direction
write the equations of the axes and coordinate planes	axes
find the angle between two lines; test perpendicular and parallel	angles
write a line in vector and Cartesian form	equations

Diagram

$OP$ makes angles $\alpha$, $\beta$, $\gamma$ with the axes: $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$ (sketch, not to scale).

If a line makes angles $\alpha$ , $\beta$ , $\gamma$ with the axes, $l = \cos\alpha$ , $m = \cos\beta$ , $n = \cos\gamma$ and $l^2 + m^2 + n^2 = 1$ .