Fun with Math!: Angle between Planes

If we have two planes, how do we find the angle between them? Take a look at the picture below. The acute angle between the planes is the same as the acure angle between their normal vectors. Therefore, we can find the angle between the normal vectors to get the angle between the planes.

Recall the angle between two vectors is

$\theta=\cos^{-1}\left ( \frac{\textbf{u}\cdot\textbf{v}}{|\textbf{u}||\textbf{v}|} \right )$

from the definition of the dot product. The formula will be modified to become

$\theta=\cos^{-1}\left ( \frac{\textbf{n}_1\cdot\textbf{n}_2}{|\textbf{n}_1||\textbf{n}_2|} \right )$

to suit the name of the vectors here.

The graph below shows 2 planes in cyan and purple respectively. Each plane contains 3 vectors accordingly. Moving the initial or terminal points of the joining vectors will move the planes. Move the normal vectors u_n and v_n only if they are unclear (this won't move the plane). The angle between the planes is stated on top of the graph. Click here if you want to rotate the graph by yourself and have clearer picture.

Betty, Created with GeoGebra and Thomas Calculus, Pearson.

Fun with Math!

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Sunday, May 10, 2020

Angle between Planes

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