Relative to the origin $O$, the position vectors of the fixed points $A$ and $B$ are \(\mathbf

Question

Relative to the origin $O$, the position vectors of the fixed points $A$ and $B$ are $\mathbf{a}$ and $\mathbf{b}$ respectively. It is given that $|\mathbf{a}|=1$, $|\mathbf{b}|=2$ and $|\mathbf{3b}-\mathbf{2a}|=\sqrt{34}$. By considering the scalar product $(3\mathbf{b}-2\mathbf{a})\cdot (3\mathbf{b}-2\mathbf{a})$, find the value of $\mathbf{a}\cdot \mathbf{b}$.[3] Hence find the exact value of $|\mathbf{a}\times \mathbf{b}|$ and give a geometrical meaning of this value.[3] A point $P$ divides $AB$ in the ratio $2:1$ and a point $M$ is the mid-point of $BC$, where the point $C$ has position vector $\mathbf{c}$. It is given that the point $N$ is such that $OBMN$ forms a rhombus and $P$ is also the mid-point of $MN$. Show that $2\mathbf{a}+4\mathbf{b}-3\mathbf{c}=\mathbf{0}$.[3]

Accepted Answer

(a)(i) $\mathbf a\cdot\mathbf b=\frac12$ (ii) $|\mathbf a\times\mathbf b|=\frac{\sqrt{15}}2$, the area of the parallelogram with adjacent sides $OA$ and $OB$, equivalently the perpendicular distance from $B$ to $OA$ (b) $2\mathbf a+4\mathbf b-3\mathbf c=\mathbf0$

2023 MI P1 Q8

2023 MI P1 Q8

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