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“Three-Dimensional Geometry in Depth: Planes Through Lines, Images and Bisectors” is available to Standard subscribers.

Beyond the textbook. Find the plane through a line and a point, the plane through two lines, and the plane in a whole family. Reflect a point in a plane or a line, find a shortest path that bounces off a mirror, and bisect the angle between two planes. It is at the level of JEE Advanced, A-Level Further Maths and IB Higher Level, with a full timed paper.

3Lessons12Problems2Python Labs1Question Set1Mock Test

Read the first lesson freePlanes through lines and points

What's inside

Planes through lines and points: Every plane through the line where two planes $P_1 = 0$ and $P_2 = 0$ meet has the form $P_1 + \lambda P_2 = 0$

The chapter itself stays free

This is an extra course for Three-Dimensional Geometry. Its main courses are open to everyone:

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Images and feet of perpendiculars: The image of a point

P

in a plane is the point

P'

on the other side, on the perpendicular from

P

, at the same distance. It is

P' = P - 2\,\dfrac{\vec{p} \cdot \vec{n} - k}{\lvert \vec{n} \rvert^2}\,\vec{n}

. The image in a line is

2F - P

, with

F

the foot of the perpendicular. A shortest path that bounces off a mirror is found by using the image.

Bisecting planes and harder distance problems: The planes that bisect the angle between

A_1x + B_1y + C_1z + D_1 = 0

and

A_2x + B_2y + C_2z + D_2 = 0

are

\dfrac{A_1x + B_1y + C_1z + D_1}{\lvert \vec{n}_1 \rvert} = \pm\dfrac{A_2x + B_2y + C_2z + D_2}{\lvert \vec{n}_2 \rvert}

. The plane equidistant from two points is the perpendicular bisector plane. A tetrahedron has volume

\dfrac{1}{6}\lvert [\overrightarrow{AB}\ \overrightarrow{AC}\ \overrightarrow{AD}] \rvert