Specialist Mathematics · Unit 2 · Circle and geometric proofs · Geometric proofs using vectors
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Let \(ABCD\) be a parallelogram with adjacent sides represented by vectors \(\vec{a}\) and \(\vec{b}\). Prove that the diagonals of the parallelogram are perpendicular if and only if \(|\vec{a}| = |\vec{b}|\).
The diagram shows parallelogram OABC with position vectors $\vec{OA} = \mathbf{a}$ and $\vec{OC} = \mathbf{c}$. Prove that the diagonals of the parallelogram meet at right angles if and only if it is a rhombus.
A quadrilateral \(PQRS\) has vertices with position vectors \(\vec{p}\), \(\vec{q}\), \(\vec{r}\), and \(\vec{s}\). The quadrilateral is a parallelogram with \(\vec{PQ} = \vec{SR}\). Prove that if the diagonals \(PR\) and \(QS\) meet at right angles, then \(|\vec{PQ}| = |\vec{PS}|\).