A force of - P ^ k acts on the origin of the coordinate system. The...

It is given that the force \overset{\to}{F} = - P\overset{^}{k}

The coordinates of the origin is $( 0, 0 )$.

The coordinates of the point given is $( 2, - 3 )$

The displacement vector is

\overset{\to}{r} = ( 0 - 2 ) \overset{^}{i} + ( 0 + 3 ) \overset{^}{j} = - 2\overset{^}{i} + 3\overset{^}{j}

Hence, the torque is

\overset{\to}{τ} = \overset{\to}{r} \times \overset{\to}{F} = ( - 2\overset{^}{i} + 3\overset{^}{j} ) \times ( - P\overset{^}{k} )

⇒ \overset{\to}{τ} = - 2P\overset{^}{j} - 3P\overset{^}{i} = P ( a\overset{^}{i} + b\overset{^}{j} )

So, it can be written

$a = - 3$, $b = - 2$

$∴ \dfrac{a}{b} = \left(\dfrac{3}{2}\right) ⇒$ so $x = 3$