Step 1: Express the given vectors in component form. Given $OP = i - 3j$, so the position vector of P is $\begin{pmatrix} 1 \\ -3 \end{pmatrix}$. Given $PQ = 4i - 2j$, so the vector from P to Q is $\begin{pmatrix} 4 \\ -2 \end{pmatrix}$. a) The coordinates of Q. To find the position vector of Q, we add the vector $OP$ and the vector $PQ$. $$OQ = OP + PQ$$ $$OQ = \begin{pmatrix} 1 \\ -3 \end{pmatrix} + \begin{pmatrix} 4 \\ -2 \end{pmatrix}$$ $$OQ = \begin{pmatrix} 1+4 \\ -3-2 \end{pmatrix}$$ $$OQ = \begin{pmatrix} 5 \\ -5 \end{pmatrix}$$ The coordinates of Q are $(5, -5)$. b) $|PQ|$ in surd form. The vector $PQ = 4i - 2j$. The magnitude of a vector $v = ai + bj$ is given by $|v| = \sqrt{a^2 + b^2}$. For $PQ = 4i - 2j$, we have $a=4$ and $b=-2$. $$|PQ| = \sqrt{4^2 + (-2)^2}$$ $$|PQ| = \sqrt{16 + 4}$$ $$|PQ| = \sqrt{20}$$ To express this in surd form, we simplify $\sqrt{20}$: $$\sqrt{20} = \sqrt{4 \times 5}$$ $$\sqrt{20} = \sqrt{4} \times \sqrt{5}$$ $$|PQ| = 2\sqrt{5}$$ The final answers are: a) The coordinates of Q are $\boxed{(5, -5)}$. b) $|PQ|$ in surd form is $\boxed{2\sqrt{5}}$. Send me the next one 📸