The position vectors of the points $A$ and relative to the origin $O$ are $\mathbf{a}$ and \(\

Question

The position vectors of the points $A$ and relative to the origin $O$ are $\mathbf{a}$ and $\mathbf{b}$ respectively. Point $M$ on ${AB}$ is such that $AM:MB=1:2$. Find $\overrightarrow{OM}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. If the area of triangle $OBM$ is $4\ \mathrm{unit}{{\mathrm{s}}^{2}}$, find $\left| \mathbf{a}\times \mathbf{b} \right|$.[5] Another point $P$ has position vector $\mathbf{p}$ and $\mathbf{p}\ne \mathbf{0}$. Given $\left( \mathbf{p}-\mathbf{a} \right)\times \left( \mathbf{b}-\mathbf{a} \right)=\mathbf{0}$, what can you deduce about the relationship between the vectors $\overrightarrow{AP}$ and $\overrightarrow{AB}$? Hence find the vector equation of line $l$ that passes through points $A$ and $P$ in terms of $\mathbf{b}$ and $\mathbf{a}$.[2] Given instead that $\mathbf{a}=\left( 4\mathbf{a}\cdot \mathbf{b} \right)\mathbf{b}$, describe in words the relation between $\mathbf{a}$ and $\mathbf{b}$ and find $\left| \mathbf{b} \right|$.[3]

Accepted Answer

(i) $\left| \mathbf{a}\times \mathbf{b} \right|=12$ (ii) $\overrightarrow{AP}\times \overrightarrow{AB}=\mathbf{0}$. $\overrightarrow{AP}$ is parallel to vector $\overrightarrow{AB}$. Since line $l$ that passes through point $A$ and is parallel to vector $\overrightarrow{AB}$, ${{l}_{AP}}:\mathbf{r}=\mathbf{a}+\lambda \left( \mathbf{b}-\mathbf{a} \right)$, $\lambda \in \mathbb{R}$. (iii) $\left| \mathbf{b} \right|=\frac{1}{2}$ since $\left| \mathbf{b} \right|>0$

2023 SAJC Promo Q5

2023 SAJC Promo Q5

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