3D rotation about an arbitrary axis (3d Math Primer)

3dlinear algebra

I am reading a 3d math primer book and I don't understand the following paragraph. Please Help me. I have been stuck on this for 2 days.

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Notice that $\mathbf w$ and ${\mathbf v}_\bot$ form a 2D coordinate space, with ${\mathbf v}_\bot$ as the “x-axis” and w as the “y-axis.” (Note that the two vectors don’t necessarily have unit length.) ${\mathbf v}'_\bot$ is the result of rotating ${\mathbf v}'$ in this plane by the angle $\theta$. Note that this is almost identical to rotating an angle into standard position. Section 1.4.4 showed that the endpoints of a unit ray rotated by an angle $\theta$ are $\cos\theta$ and $\sin\theta$. The only difference here is that our ray is not a unit ray, and we are using ${\mathbf v}_\bot$ and $\mathbf w$ as our basis vectors. Thus, ${\mathbf v}'_\bot$ can be computed as

$${\mathbf v}'_\bot = \cos(\theta) {\mathbf v}_\bot + \sin(\theta){\mathbf w}$$

Can someone please help how he got ${\mathbf v}'_\bot$. AFIK the coordinates of ${\mathbf v}'_\bot$ will be

$$\begin{pmatrix}|{\mathbf v}'_\bot| \cdot \cos\theta\\|{\mathbf v}'_\bot| \cdot \sin\theta\end{pmatrix}$$

A step by step explanation will be highly appreciated. Thank you.

Best Answer

  1. First of all we note that ${\mathbf w}=\hat{\mathbf n}\times {\mathbf v}_{\bot}$. This is important because $$ \|{\mathbf w}\|=\|\hat {\mathbf n}\|\cdot \|{\mathbf v}_{\bot}\| \cdot \sin\frac{\pi}2 = \|{\mathbf v}_{\bot}\|. $$
  2. Notice that the unit vectors $\frac{{\mathbf v}_{\bot}}{\|{\mathbf v}_{\bot}\|}$ and $\frac{{\mathbf w}}{\|{\mathbf w}\|}$ form an orthonormal basis. According to the results from the Section 1.4.4, ${\mathbf v}_{\bot}'$ in this basis has the coordinates $(\|{\mathbf v}_{\bot}\|\cos\theta,\|{\mathbf v}_{\bot}\|\sin\theta)$, thus, it can be expressed as $$ {\mathbf v}_{\bot}'=\|{\mathbf v}_{\bot}\|\cos\theta \cdot\frac{{\mathbf v}_{\bot}}{\|{\mathbf v}_{\bot}\|}+\|{\mathbf v}_{\bot}\|\sin\theta\cdot \frac{{\mathbf w}}{\|{\mathbf w}\|} ={\mathbf v}_{\bot}\cos\theta+{\mathbf w}\sin\theta. $$
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