A force of $$-P\hat{k}$$ acts on the origin of the coordinate system. The torque about the point (2, -3) is $$P(a\hat{i} + b\hat{j})$$. The ratio of $$\frac{a}{b}$$ is $$\frac{x}{2}$$. The value of x is _______

Force $$\vec{F} = -P\hat{k}$$ acts at origin. Torque about point (2,-3,0):

$$\vec{r} = (0,0,0) - (2,-3,0) = (-2, 3, 0)$$

$$\vec{\tau} = \vec{r} \times \vec{F} = (-2,3,0) \times (0,0,-P)$$

$$= (3(-P)-0, 0-(-2)(-P), 0) = (-3P, -2P, 0) = P(-3\hat{i} - 2\hat{j})$$

Given $$\vec{\tau} = P(a\hat{i} + b\hat{j})$$: $$a = -3, b = -2$$. $$a/b = 3/2 = x/2$$, so $$x = 3$$.

The answer is 3.