Given that $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors such that \(\mathbf{a}=(

Question

Given that $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors such that $\mathbf{a}=(\mathbf{a}\cdot\mathbf{b})\mathbf{b}$. State the relationship between $\mathbf{a}$ and $\mathbf{b}$. Find $|\mathbf{b}|$.[2] With reference to the origin $O$, the points $M$ and $N$ are such that $\overrightarrow{OM}=\alpha\mathbf{i}+4\mathbf{j}+\beta\mathbf{k}$ and $\overrightarrow{ON}=4\mathbf{i}+\alpha\mathbf{j}+\beta\mathbf{k}$, where $\alpha$ and $\beta$ are constants. Given that $M$ lies in the $xy$-plane and $|\overrightarrow{OM}|=5$, find the values of $\alpha$ and $\beta$.[3] Given instead that $\overrightarrow{OM}$ is perpendicular to $\overrightarrow{ON}$, find an expression for $|\overrightarrow{OM}\times\overrightarrow{ON}|$ in terms of $\alpha$, giving your answer in fully factorised form. [3]

Accepted Answer

(a) parallel, $|\mathbf b|=1$ (b)(i) $\alpha=\pm3$, $\beta=0$ (b)(ii) $(\alpha-4)^2$

2025 NYJC Promo Q7

2025 NYJC Promo Q7

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