Find the sum of the vectors 3i + 7j and 2i – 2j. If the position vectors of P and Q are i + 3j and 5i – 2j respectively, find PQ and |PQ|.

$$(3+2)\mathbf{i} + (7-2)\mathbf{j} = 5\mathbf{i} + 5\mathbf{j}$$

The sum of the vectors is \boxed{\text{5\mathbf{i} + 5\mathbf{j}}}

3. If the position vectors of P and Q are $\mathbf{i} + 3\mathbf{j}$ and $5\mathbf{i} - 2\mathbf{j}$ respectively, find $\vec{PQ}$ and $|\vec{PQ}|$.

Let $\vec{P} = \mathbf{i} + 3\mathbf{j}$ and $\vec{Q} = 5\mathbf{i} - 2\mathbf{j}$.

$$\vec{PQ} = \vec{Q} - \vec{P} = (5\mathbf{i} - 2\mathbf{j}) - (\mathbf{i} + 3\mathbf{j})$$

If $\vec{PQ} = x\mathbf{i} + y\mathbf{j}$, then $|\vec{PQ}| = \sqrt{x^2 + y^2}$.
Here, $x=4$ and $y=-5$.

$$|\vec{PQ}| = \sqrt{(4)^2 + (-5)^2}$$

$$|\vec{PQ}| = \sqrt{16 + 25} = \sqrt{41}$$

The vector $\vec{PQ}$ is \boxed{\text{4\mathbf{i} - 5\mathbf{j}}} and its magnitude $|\vec{PQ}|$ is \boxed{\text{\sqrt{41}}}.