Relative to the origin $O$, the points $A$ and $C$ have position vectors $\mathbf{a}$ and \(

Question

Relative to the origin $O$, the points $A$ and $C$ have position vectors $\mathbf{a}$ and $2\mathbf{b}$ such that $$\mathbf{a}=5p\mathbf{i}-2p\mathbf{j}+4p\mathbf{k} \text{ and}\hspace{0.5em} \mathbf{b}=\mathbf{i}-2\mathbf{j}+2\mathbf{k},$$ where $p$ is a positive constant. It is given that $|\mathbf{a}|=2|\mathbf{b}|$. Find the exact value of $p$.[2] Evaluate $(\mathbf{a}+2\mathbf{b}) \cdot (\mathbf{a}-2\mathbf{b})$.[1] Evaluate $\mathbf{a} \times \mathbf{b}$ and hence find the area of triangle $OAC$.[3] Use a geometrical reason to explain why $|\mathbf{a} \times \mathbf{b}| = \frac{1}{4}(|\mathbf{a}+2\mathbf{b}|)(|\mathbf{a}-2\mathbf{b}|)$.[2]

Accepted Answer

(a) $p=\dfrac{2\sqrt5}{5}$ (b) $0$ (c) $\mathbf a\times\mathbf b=\dfrac{4\sqrt5}{5}\begin{pmatrix}2\-3\-4\end{pmatrix}$, area $=\dfrac45\sqrt{145}\text{ units}^2$ (d) $|\mathbf a\times\mathbf b|=\dfrac14|\mathbf a+2\mathbf b||\mathbf a-2\mathbf b|$

2025 RI P1 Q3

2025 RI P1 Q3

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