If the magnitude of the moment about the point $\hat j +\hat k$ of a force $\hat i +α\hat j-\hat k$ acting through the point $\hat i + \hat j$ is $\sqrt{8}$, then the value of $α$, is

2

Let $\vec M$ be the moment of the force $\vec F = \hat i +α\hat j-\hat k$ acting through the point $A (\hat i +\hat j)$ about the point $O(\hat j +\hat k)$.

Then,

$\vec M=\vec{OA}×\vec F$

$⇒\vec M=(\hat i-\hat k)×(\hat i +α\hat j-\hat k)$

$⇒\vec M=\begin{vmatrix}\hat i&\hat j&\hat k\\1&0&-1\\1&α&-1\end{vmatrix}=α\hat i+α\hat k$

$⇒|\vec M|=\sqrt{α^2+α^2}=\sqrt{2}|α|$

$⇒\sqrt{2}|α|=\sqrt{8}$

$⇒|α|=2⇒α=±2$