Three Dimensional Geometry — Part 1: Direction cosines and the equation of a line: visual lesson, CBSE Class 12 Mathematics

Lesson

Three Dimensional Geometry, Part 1

Lines in space: which way they point, and how to write them down.

In this lesson you will

Find direction cosines and direction ratios of a line
Use the fact that the squares of the direction cosines add to 1
Test points for collinearity using direction ratios
Write the vector and Cartesian equations of a line
Convert a Cartesian equation to standard form

Two points fix a line in space (sketch, not to scale).

Concept

What you will be able to do

Find direction cosines and direction ratios
Use $l^2 + m^2 + n^2 = 1$
Test collinearity of three points
Write the vector and Cartesian equations of a line
Rewrite a line in standard form

Concept

Direction cosines

If a directed line makes angles $\alpha$ , $\beta$ , $\gamma$ with the positive $x$ -, $y$ - and $z$ -axes, its direction cosines are $l = \cos\alpha$ , $m = \cos\beta$ , $n = \cos\gamma$ .

l = \cos\alpha, \quad m = \cos\beta, \quad n = \cos\gamma

$OP$ makes angles $\alpha$, $\beta$, $\gamma$ with the axes; $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$.

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