Explanation

(a)
Let the point $P$ be $(x,\ y,\ z)$, then the vector $\left( x\widehat{i} + y\widehat{j} + z\widehat{k} \right)$ will lie on the line
$\Rightarrow (x - 1)\widehat{i} + (y - 1)\widehat{j} + (z - 1)\widehat{k} = - \lambda\widehat{i} + \lambda\widehat{j} - \lambda\widehat{k}$
$\Rightarrow x = 1 - \lambda,\ y = 1 + \lambda\ \$and $\ z = 1 - \lambda$
Now point $P$ is nearest to the origin $\Rightarrow D = (1 - \lambda)^{2} + (1 + \lambda)^{2} + (1 - \lambda)^{2}$
$\Rightarrow \frac{dD}{d} = - 4(1 - \lambda) + 2(1 + \lambda) = 0\ \Rightarrow \ \lambda = \frac{1}{3}$
$\Rightarrow$ The point is $\left( \frac{2}{3},\frac{4}{3},\frac{2}{3} \right)$