Find the value of $k$, so that the lines $x = -y = kz$ and $x-2 = 2y+1 = -z+1$ are perpendicular to each other.

$2$

The correct answer is Option (3) → $2$ ##

$\frac{x}{k} = \frac{y}{-k} = \frac{z}{1} \quad \dots \text{line (i)}$

$\frac{x-2}{1} = \frac{y+1/2}{1/2} = \frac{z-1}{-1} \quad \dots \text{line (ii)}$

$\text{dR of line (i)} = \langle k, -k, 1 \rangle$

$\text{dR of line (ii)} = \langle 1, 1/2, -1 \rangle$

As they are perpendicular:

$k - \frac{k}{2} - 1 = 0$

$\frac{k}{2} = 1 \Rightarrow k = 2$