Skew Lines and Shortest Distances

Name: Skew Lines and Shortest Distances
Author: StemPy Lab

Two cables hang across a street at different heights. They do not touch, and they are not parallel. How close do they come? Such lines are called skew, and they exist only in space. Test whether two lines meet, find the shortest distance between two lines, and find the distance of a point from a line together with the foot of the perpendicular. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level Further Maths and IB.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Coplanar lines and skew lines
Two lines in space may be parallel, may meet, or may do neither. Lines that are not parallel and never meet are called skew. Two lines are coplanar when they lie in one plane, and the test is that the scalar triple product $(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)$ is $0$ .
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 2about 25 min
The shortest distance between two lines
The shortest distance between two skew lines is measured along their common perpendicular. Its direction is $\vec{b}_1 \times \vec{b}_2$ , so the distance is the length of the part of $\vec{a}_2 - \vec{a}_1$ along that direction: $d = \dfrac{\lvert (\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) \rvert}{\lvert \vec{b}_1 \times \vec{b}_2 \rvert}$ .
1 simulation1 practice question1 Python program4 problems
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Lesson 3about 25 min
Distance of a point from a line and the foot of the perpendicular
The distance of a point $P$ from a line through $A$ with direction $\vec{b}$ is $\dfrac{\lvert (\vec{P} - \vec{A}) \times \vec{b} \rvert}{\lvert \vec{b} \rvert}$ . The foot of the perpendicular is the point $F = A + t\vec{b}$ with $t = \dfrac{(P - A) \cdot \vec{b}}{\lvert \vec{b} \rvert^2}$ , and the distance is $PF$ .
1 simulation1 practice question1 Python program3 problems1 Python lab1 question set1 note
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Skew Lines and Distances: Timed Test

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Skew Lines and Shortest Distances

Course outline (3 lessons)

Coplanar lines and skew lines

The shortest distance between two lines

Distance of a point from a line and the foot of the perpendicular

Test Your Learning

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