A particle moves from a point P to a point Q . The coordinates of...

The position vector of point $P$ is $\vec{r}_P = 3\hat{i} - 2\hat{j} + 4\hat{k}$.

The displacement vector from $P$ to $Q$ is given by $\vec{D} = \vec{r}_Q - \vec{r}_P$.

Rearranging this equation, the position vector of point $Q$ is:
$\vec{r}_Q = \vec{r}_P + \vec{D}$
$\Rightarrow \vec{r}_Q = (3\hat{i} - 2\hat{j} + 4\hat{k}) + (2\hat{i} - 5\hat{j} + 4\hat{k})$
$\Rightarrow \vec{r}_Q = 5\hat{i} - 7\hat{j} + 8\hat{k}$

The distance of point $Q$ from the origin is the magnitude of its position vector:
$|\vec{r}_Q| = \sqrt{5^2 + (-7)^2 + 8^2}$
$|\vec{r}_Q| = \sqrt{25 + 49 + 64} = \sqrt{138}$

Answer: $\sqrt{138}$