The non-zero vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are such that \(\mathbf{

Question

The non-zero vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are such that $\mathbf{a} \times \mathbf{b} = \mathbf{c} \times \mathbf{a}$. Given that $\mathbf{b} \neq -\mathbf{c}$, find a linear relationship between $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$.[3] The variable position vector $\mathbf{v}$ satisfies the equation $\mathbf{v} \times (\mathbf{i} - 3\mathbf{k}) = 2\mathbf{j}$. Find the set of position vectors $\mathbf{v}$.[3] Describe the set of points represented by position vectors $\mathbf{v}$ geometrically.[2]

Accepted Answer

(a) $\mathbf a=\lambda(\mathbf b+\mathbf c)$, $\lambda\ne0$ (b)(i) $V=\left\{\begin{pmatrix}0\0\2\end{pmatrix}+t\begin{pmatrix}1\0\-3\end{pmatrix}:t\in\mathbb R\right\}$ (ii) A line through $(0,0,2)$ parallel to $\begin{pmatrix}1\0\-3\end{pmatrix}$

2025 Specimen Paper P1 Q6

2025 Specimen Paper P1 Q6

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