Let the line L : x - 1 2 = y + 1 - 1 = z - 3 1 intersect the plane 2 x + y + 3 z = 16 at the point P . Let…

We have,

$L:\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{1}=\lambda$

Any point on $L$ is $P\left(\left(2\lambda +1\right),\left(-\lambda -1\right),\left(\lambda +3\right)\right)$

It lies on plane $2x+y+3z=16$, so

$\Rightarrow 2\left(2\lambda +1\right)+\left(-\lambda -1\right)+3\left(\lambda +3\right)=16$

$\Rightarrow 6\lambda +10=16\Rightarrow \lambda =1$

$\therefore P\equiv \left(3,-2,4\right)$

Let $Q\left(\left(2\mu +1\right),\left(-\mu -1\right),\left(\mu +3\right)\right)$

Direction ratios of $QR$ is

 

Direction ratios of $L$ is .

Since, $QR\perp L$, so

$4\mu +\mu +\mu +6=0$

$\Rightarrow \mu =-1$

$Q\equiv \left(-1,0,2\right)$

Now,

$$\begin{gathered} \overrightarrow{QR}=2\hat{i}-\hat{j}-5\hat{k} \\ \overrightarrow{QP}=4\hat{i}-2\hat{j}+2\hat{k} \end{gathered}$$

Hence,

$\overrightarrow{QR}\times \overrightarrow{QP}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& -1& -5\\ 4& -2& 2\end{array}\right|=-12\hat{i}-24\hat{j}$

Therefore,

$\alpha =\frac{1}{2}\times \sqrt{144+576}$

$\Rightarrow {\alpha }^2=\frac{720}{4}=180$