Relative to the origin $O$, two points $A$ and $B$ have position vectors given by \(\mathbf{a}

Question

Relative to the origin $O$, two points $A$ and $B$ have position vectors given by $\mathbf{a}=2\mathbf{i}-\mathbf{j}$ and $\mathbf{b}=\alpha \mathbf{k}$ respectively, where $\alpha $ is a negative real constant. Find the angle between $OA$ and $OB$.[1] Let $\angle OBA=\theta $, where $\theta $ is in degrees. It is given that $\sin \theta =\frac{1}{2}$. Find the two possible values of $\theta $, and justify which of the two values is the correct value of $\theta $.[2] By considering a suitable scalar product, find the exact value of $\alpha $.[3] A point $C$ has position vector $\mathbf{c}$ such that $\mathbf{c}=\mathbf{b}-\mathbf{a}$. State the shape of the quadrilateral $OABC$, justifying your answer. Find the exact area of the quadrilateral $OABC$.[3]

Accepted Answer

(a) $90^\circ$ (b) Possible values $30^\circ$ and $150^\circ$, with $\theta=30^\circ$ because it is acute (c) $\alpha=-\sqrt{15}$ (d) Parallelogram with area $5\sqrt3$ square units

2023 HCI P1 Q8

2023 HCI P1 Q8

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