Relative to the origin, the points $A$, $B$,$C$ and $D$ have position vectors$2\mathbf{i}+3\mathbf{j

Question

Relative to the origin, the points $A$, $B$,$C$ and $D$ have position vectors$2\mathbf{i}+3\mathbf{j}-4\mathbf{k}$, $3\mathbf{i}+4\mathbf{j}-4\mathbf{k}$, $4\mathbf{i}+p\mathbf{j}$ and $q\mathbf{i}+\mathbf{j}-\mathbf{k}$ respectively, where $p$ and $q$ are constants. Given that $AC$ is perpendicular to both $AB$ and $DC$, show that $p=1$ and $q=6$.[3] The point $E$ is on $AC$ produced such that $2AC=CE$ and $l$ is the line through $E$ perpendicular to both $AB$ and $AE$. Find the coordinates of $E$ and hence find the vector equation of $l$.[4] the position vectors of the two points on $l$ which are $\sqrt{243}$ units from $A$.[4]

Accepted Answer

$p=1,\ q=6$ (i) $E=(8,-3,8)$, $l:\mathbf r=\begin{pmatrix}8\-3\8\end{pmatrix}+\lambda\begin{pmatrix}-1\1\1\end{pmatrix},\ \lambda\in\mathbb R$ (ii) $(5,0,11)$ and $(11,-6,5)$

2013 PJC P1 Q9

2013 PJC P1 Q9

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