The points $P$ and $Q$ have position vectors $\mathbf{a}+\mathbf{b}$ and $3\mathbf{a}-2\mathbf{b}$ r

Question

The points $P$ and $Q$ have position vectors $\mathbf{a}+\mathbf{b}$ and $3\mathbf{a}-2\mathbf{b}$ respectively relative to the origin $O$. Given that $OPQR$ is a parallelogram, express the vectors $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. By evaluating two scalar products, show that if $OPQR$ is a square, then ${{\left| \mathbf{a} \right|}^{2}}=2{{\left| \mathbf{b} \right|}^{2}}$.

Accepted Answer

$\overrightarrow{PQ}=2\mathbf a-3\mathbf b$, $\overrightarrow{PR}=\mathbf a-4\mathbf b$ If $OPQR$ is a square, $|\mathbf a|^2=2|\mathbf b|^2$

Parallelogram and Square

Parallelogram and Square

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