Let the line L: (x - 1)/(2) = (y + 1)/(- 1) = (z - 3)/(1) intersect...

We have,

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

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

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

$⇒ 6\lambda + 10 = 16 ⇒ \lambda = 1$

$∴ P ≡ \left(3, - 2, 4\right)$

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

Direction ratios of $QR$ is

\left<2\mu, - \mu, \mu + 6\right>$</p><p>Direction ratios of$L$is \left<2, - 1, 1\right>$.

Since, $QR ⊥ L$, so

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

$⇒ \mu = - 1$

$Q ≡ \left(- 1, 0, 2\right)$

Now,

\overset{\to}{QR} = 2\overset{^}{i} - \overset{^}{j} - 5\overset{^}{k}\overset{\to}{QP} = 4\overset{^}{i} - 2\overset{^}{j} + 2\overset{^}{k}

Hence,

\overset{\to}{QR} \times \overset{\to}{QP} = \left|\begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ 2 & - 1 & - 5 \\ 4 & - 2 & 2\end{cases}\right| = - 12\overset{^}{i} - 24\overset{^}{j}

Therefore,

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

$⇒ {\alpha}^{2} = \dfrac{720}{4} = 180$