Explanation

(a)
Let the line joining the points with position vectors $- 2\widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} + 5\widehat{\mathbf{k}}$ and $7\widehat{\dot{\mathbf{i}}} - \widehat{\mathbf{k}}$ be
Divide in the ratio λ:1 by $\widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}}$
$\therefore\frac{\lambda\left( 7\widehat{\dot{\mathbf{i}}} - \widehat{\mathbf{k}} \right) + ( - 2\widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} + 5\widehat{\mathbf{k}})}{\lambda + 1} = \widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}}$
$\Longrightarrow (7\lambda - 2)\widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} + (5 - \lambda)\widehat{\mathbf{k}} = (\lambda + 1)\widehat{\dot{\mathbf{i}}} + 2(\lambda + 1)\widehat{\dot{\mathbf{j}}} + 3(\lambda + 1)\widehat{\mathbf{k}}$
On equating the coefficient of $\widehat{\dot{\mathbf{i}}}$ , we get
$7\lambda - 2 = \lambda + 1 \Longrightarrow \lambda = 2$
Hence, required ratio $= \lambda:1 = 2:1$