Making plane parallel to $xy$ axis

Question

I want to rotate plane: $2x + 4y + 2z + 4 = 0$ to make it parallel to $xy$ axis by using rotation matrix. In specific, how to calculate angles of rotation? I can't find any good explanation on the web, so if anyone knows how to do it, please help. Thanks.

Answer 1

Your plane has normal vector $(2,4,2)$ as read from the coefficients of the equation. You want to rotate that to some multiple of $(0,0,1)$.

how to calculate angles of rotation

You can use the fact that the inner product (dot product) is proportional to the cosine of the angle between the vectors:

$$\langle v,w\rangle=\lVert v\rVert\cdot\lVert w\rVert\cdot\cos\theta$$

which you can rearrange to

$$\theta=\arccos\frac{\langle v,w\rangle}{\lVert v\rVert\cdot\lVert w\rVert}$$

giving you an angle in the range $0°\le\theta\le180°$ as expected. In your case:

$$\theta=\arccos\frac{\left\langle\begin{pmatrix}2\\4\\2\end{pmatrix},\begin{pmatrix}0\\0\\1\end{pmatrix}\right\rangle}{\left\lVert\begin{pmatrix}2\\4\\2\end{pmatrix}\right\rVert\cdot\left\lVert\begin{pmatrix}0\\0\\1\end{pmatrix}\right\rVert}=\arccos\frac{2}{\sqrt{24}\cdot1}=\arccos\frac1{\sqrt6}$$

Usually you'd need to combine this rotation angle with a definition of the axis of rotation. That axis direction would be orthogonal to both the vectors I gave, so you can use the cross product of these vectors to compute the axis.