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

Question

Given that $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors such that $\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{a}$, find the relationship between $\mathbf{a}$ and $\mathbf{b}$.[2] The points $P$,$Q$ and $R$ have position vectors $\mathbf{p}$, $\mathbf{q}$ and $\mathbf{r}$ respectively. The points $P$ and $Q$ are fixed and $R$ varies.[3] Given that $\mathbf{q}$ is non-zero and $(\mathbf{r}-\mathbf{p})\times \mathbf{q}=\mathbf{0}$, describe geometrically the set of all possible positions of the point $R$.[3] Given instead that $\mathbf{r}=\left( \begin{matrix}x \y \z \end{matrix} \right)$, $\mathbf{p}=\left( \begin{matrix}-1 \2 \4 \end{matrix} \right)$, $\mathbf{q}=\left( \begin{matrix}3 \-5 \2 \end{matrix} \right)$ and that $(\mathbf{r}-\mathbf{p})\cdot \mathbf{q}=0$, find the relationship between $x$, $y$ and $z$. Describe the set of all possible positions of the point $R$ in this case.[4]

Accepted Answer

(a) ${\mathbf{a}}$ is parallel to ${\mathbf{b}}$ . (b)(i) ${H}$ is a line which passes thought point ${P}$ and parallel to the vector ${\mathbf{q}}$ . (ii) Point ${R}$ is on the plane which has a normal vector ${\begin{pmatrix} 3 \ -5 \ 2 \end{pmatrix}}$ and contains point ${P}$.

N2020 P1 Q5

N2020 P1 Q5

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