When I use rotation matrix z- position doesn't move. why?

Question

I'm trying to rotate 3 points around x- axis.

and three points are (1.33, 2.49, 0),(2.5, 4.33, 0),(0, 5, 0). which is on the xy plane

rotation angle is pi/4. then, I followed this cacultation.

but I think z position should not be zero and x position should be fixed as it rotated around x-axis, right? but the result is different. Could you give me any comment about this? Is there anything I missing now?

Answer 1

Are you trying to do three points at once? That's not how matrix multiplication works. You appear to have a rotation matrix given by

$$M=\begin{bmatrix}1&0&0\\0&\sqrt2&-\sqrt2\\0&\sqrt2&\sqrt2\end{bmatrix}$$ and you want to use that to rotate your point. So do your matrix multiplication on the first point: $$\begin{bmatrix}1&0&0\\0&\sqrt2&-\sqrt2\\0&\sqrt2&\sqrt2\end{bmatrix} \begin{bmatrix}1.33\\2.49\\0\end{bmatrix} = \begin{bmatrix}1\times1.33+0\times2.49+0\times2.49\\ 0\times1.33+\sqrt2\times2.49-\sqrt2\times0\\ 0\times1.33+\sqrt2\times2.49+\sqrt2\times0\end{bmatrix}$$ by taking each member of a row times each member of a column and adding. Then you can do the same for the other points.

EDIT based on comment below: And if you do want to do them all at once, you need to put each point as a column in your right-hand matrix, not a row. Apologies for the confusion.

Answer 2

The problem is that you are trying to rotate three points at once; however, the matrix of vectors that you have constructed is incorrect, instead of your current matrix:

$$\mathbf{x}^{T}=\begin{pmatrix}1.33 & 2.49 & 0 \\ 2.5 & 4.33 & 0 \\ 0 & 5 & 0\end{pmatrix}$$

You need:

$$\mathbf{x}=\begin{pmatrix}1.33 & 2.5 & 0 \\ 2.49 & 4.33 & 5 \\ 0 & 0 & 0 \end{pmatrix}$$

If you then multiply by your rotation matrix $\mathbf{R}(\frac{\pi}{4})=\left(\begin{smallmatrix}1 & 0 & 0 \\ 0 & \sqrt{2} & -\sqrt{2} \\ 0 & \sqrt{2} & \sqrt{2}\end{smallmatrix}\right)$:

$$\mathbf{x}'=\mathbf{R}\mathbf{x}=\begin{pmatrix}1.33 & 2.5 & 0 \\ 3.52 & 6.12 & 7.07 \\ 3.52 & 6.12 & 7.07\end{pmatrix}$$

In general if you have vectors $\vec{v}_{1},\dots,\vec{v}_{m}\in\mathbb{C}^{n}$, in order to rotate them using a linear map $\mathbf{M}\in\mathbb{C}^{n\times n}$ you need to construct the matrix:

$$\mathbf{x}=\begin{pmatrix}\uparrow & & \uparrow \\ \vec{v}_{1} & \cdots & \vec{v}_{m} \\ \downarrow & & \downarrow\end{pmatrix}$$

And then compute the product $\mathbf{M}\mathbf{x}$ to get your matrix where the columns are your transformed vectors.