The given vector is V = ai + bj + ck

a=0, b=1, c=1

also $|\overline{V}|=\sqrt{a^{2}+b^{2}+c^{2}}$

Step 1: Rotation of vector V about X-axis by an angle $\theta =+\theta_1$ {+ve indicates CCW}

Now, $, \mathrm{V}_{1}=\mathrm{b}_{j}+\mathrm{ck}$

Also,

$1\left(\mathrm{V}_{1}\right)=\sqrt{\mathrm{b}^{2}+\mathrm{c}^{2}}$

From figure, $\sin \theta_{1}=\mathrm{b} / \sqrt{\mathrm{b}^{2}+\mathrm{c}^{2}}$

$\cos \theta_{1}=\mathrm{c} / \sqrt{\mathrm{b}^{2}+\mathrm{c}^{2}}$

Let, $\sqrt{b^{2}+c^{2}}=\lambda$

So, $\sin \theta_{1}=b / \lambda$ and $\cos \theta_{1}=c / \lambda$

Now, the rotational matrix about x-axis is,

$[\mathrm{Rx}]=\left[\begin{array}{cccc}{1} & {0} & {0} & {0} \\ {0} & {\cos \theta} & {\sin \theta} & {0} \\ {0} & {-\sin \theta} & {\cos \theta} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right]$

On substituting the values of $\sin \theta_{1}$ and $\cos \theta_{1},$ we get,

$[\mathrm{Rx}]=\left[\begin{array}{cccc}{1} & {0} & {0} & {0} \\ {0} & {\frac{c}{\lambda}} & {\frac{b}{\lambda}} & {0} \\ {0} & {\frac{-b}{\lambda}} & {\frac{c}{\lambda}} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right]$

Step 2: Rotate vector $\mathrm{V}^{1}$ about $\mathrm{Y}$ -axis by an angle of $\theta=-\theta_{2}$ So as to consider with z-axis.

From fig, $\sin \theta_{2}=a / \sqrt{a^{2}+b^{2}+c^{2}}=a /|V|$

$\cos \theta_{2}=\sqrt{b^{2}+c^{2}} / \sqrt{a^{2}+b^{2}+c^{2}}=\lambda /|V|$

Now, rotational matrix about y-axis is,

$[\mathrm{Ry}]=\left[\begin{array}{cccc}{\cos \theta} & {0} & {-\sin (-\theta)} & {0} \\ {0} & {1} & {0} & {0} \\ {\sin (-\theta)} & {0} & {\cos (-\theta)} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right]$

$[R y]=\left[\begin{array}{cccc}{\frac{\lambda}{|\overline V|}} & {0} & {-(\frac{-a}{|\overline V|})} &0 \\ {0} & {1} & {0} & 0 \\ {\frac{-a}{|\overline V|}} & {0} & {\frac{\lambda}{|\overline V|}} & 0\\ {0} & {0} & {0} & {1}\end{array}\right]$

$[R y]=\left[\begin{array}{cccc}{\frac{\lambda}{|\overline V|}} & {0} & {\frac{a}{|\overline V|}} &0 \\ {0} & {1} & {0} & 0 \\ {\frac{-a}{|\overline V|}} & {0} & {\frac{\lambda}{|\overline V|}} & 0\\ {0} & {0} & {0} & {1}\end{array}\right]$

Now, combined transformation,

[Av] = [Rx][Ry]

$[\mathrm{Rx}]=\left[\begin{array}{cccc}{1} & {0} & {0} & {0} \\ {0} & {\frac{c}{\lambda}} & {\frac{b}{\lambda}} & {0} \\ {0} & {\frac{-b}{\lambda}} & {\frac{c}{\lambda}} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right] \left[\begin{array}{cccc}{\frac{\lambda}{|\overline V|}} & {0} & {\frac{a}{|\overline V|}} &0 \\ {0} & {1} & {0} & 0 \\ {\frac{-a}{|\overline V|}} & {0} & {\frac{\lambda}{|\overline V|}} & 0\\ {0} & {0} & {0} & {1}\end{array}\right]$

$[A v]=\left[\begin{array}{cccc}{\frac{\lambda}{|\overline V|}} & {0} & {\frac{a}{|\overline V|}} &0 \\ {\frac{-ab}{\lambda |\overline V|}} & {\frac{c}{\lambda}} & {\frac{b}{|\overline V|}} & 0 \\ {\frac{-ac}{\lambda |\overline V|}} & {\frac{-b}{\lambda}} & {\frac{c}{|\overline V|}} & 0\\ {0} & {0} & {0} & {1}\end{array}\right]$

This is the matrix [Av] which aligns a given vector v=ai+bj+ck with the vector k along positive z-axis