Two distinct points $A$ and $B$ have non-zero position vectors $\mathbf{a}$ and $\mathbf{b}$

Question

Two distinct points $A$ and $B$ have non-zero position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. The point $R$, with position vector $\mathbf{r}$, varies such that $\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a}$. Show that $\mathbf{r}-\mathbf{b}=k\,\mathbf{a}$ where $k$ is a real number.[2] Describe geometrically the set of all possible positions of the point $R$.[2] Given further that $\angle AOB={{90}^{\circ }}$, show that $\left| \mathbf{r}\cdot \mathbf{\hat{b}} \right|=\left| \mathbf{b} \right|$.[4]

Accepted Answer

(i) $\mathbf r-\mathbf b=k\mathbf a$; (ii) line through $B$ parallel to $OA$; (iii) $\left|\mathbf r\cdot\frac{\mathbf b}{|\mathbf b|}\right|=|\mathbf b|$

2022 SAJC Promo Q9

2022 SAJC Promo Q9

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