With respect to the origin $O$, the position vectors of the points $A$ and $B$ are \(-3\mathbf

Question

With respect to the origin $O$, the position vectors of the points $A$ and $B$ are $-3\mathbf{i}+2\mathbf{j}+m\mathbf{k}$ and $-12\mathbf{i}+8\mathbf{j}+4\mathbf{k}$ respectively, where $m<0$. The point $P$ is on line $AB$ such that $AP:PB=2:1$. Find the coordinates of point $P$ in terms of $m$.[2] It is given that the area of triangle $OAP$ is $4\sqrt{13}$ units$^{2}$. Show that $m=-2$.[4] Hence, or otherwise, the shortest distance from $O$ to the line $AP$.[3]

Accepted Answer

(i) $\left(-9,6,\dfrac{m+8}{3}\right)$ (ii) $m=-2$ (iii) $3.50$ units

2013 SAJC P1 Q6

2013 SAJC P1 Q6

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