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

Question

Relative to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. The point $C$ is such that $OABC$ forms a trapezium in which $OA$ and $CB$ are the pair of parallel sides of different lengths. It is given that $\overrightarrow{CB}=k\mathbf{a}$ and $M$ is the midpoint of $OB$. By evaluating the vector product of a suitable pair of vectors, explain whether it is possible for $A$, $M$ and $C$ to be collinear.[6] Write down the height of the trapezium $OABC$ in terms of $\mathbf{a}$ and $\mathbf{b}$.[1]

Accepted Answer

(i) $\overrightarrow{AM}\times\overrightarrow{AC}=\dfrac{k-1}{2}(\mathbf{a}\times\mathbf{b})$, which is $\mathbf{0}$ only if $k=1$; then $OA=CB$, contradicting different lengths, so not possible. (ii) $\dfrac{\lvert\mathbf{a}\times\mathbf{b}\rvert}{\lvert\mathbf{a}\rvert}$

2026 NJC Promo Q4

2026 NJC Promo Q4

Solution:

Solution locked