Three Dimensional Geometry — Part 2: Angles between lines and shortest distance: visual lesson, CBSE Class 12 Mathematics

Lesson

Three Dimensional Geometry, Part 2

When lines miss each other: angles and shortest distances.

In this lesson you will

Skew lines and their common perpendicular $MN$.

Concept

Concept

For lines along $\vec{b}_1$ and $\vec{b}_2$ (ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ ), the acute angle $\theta$ satisfies

\cos\theta = \left|\dfrac{\vec{b}_1 \cdot \vec{b}_2}{|\vec{b}_1|\,|\vec{b}_2|}\right| = \dfrac{|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}}

Only the directions matter for angles.
Condition	Meaning
$\cos\theta = \dfrac{\|a_1a_2 + b_1b_2 + c_1c_2\|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\,\sqrt{a_2^2 + b_2^2 + c_2^2}}$	acute angle between the lines
$a_1a_2 + b_1b_2 + c_1c_2 = 0$	perpendicular
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$	parallel (or coincident)