The position vectors of the points $A$, $B$, $C$ and $D$ are given by $\overrightarrow{OA}=2\mathbf{

Question

The position vectors of the points $A$, $B$, $C$ and $D$ are given by $\overrightarrow{OA}=2\mathbf{i}+4\mathbf{j}-4\mathbf{k}$, $\overrightarrow{OB}=4\mathbf{i}+5\mathbf{j}$, $\overrightarrow{OC}=-\mathbf{i}-2\mathbf{j}-\mathbf{k}$ and $\overrightarrow{OD}=2\mathbf{i}-\mathbf{j}+4\mathbf{k}$ where $O$ is the origin. Show that $AC$ is perpendicular to both $AB$ and $DC$. [3] The point $E$ on $AC$ is such that $2\overrightarrow{AE}=\overrightarrow{EC}$ and $l$ is the line through $E$ perpendicular to $AB$ and to $AE$. Find a vector parallel to $l$. [3] Find the position vectors of the two points on $l$ which are a distance of $\sqrt{20}$ units from $A$. [4]

Accepted Answer

AC ⊥ AB and DC; (i) $\mathbf{d} = \begin{pmatrix} -3 \\ 2 \\ 1 \end{pmatrix}$; (ii) $\begin{pmatrix} -2 \\ 4 \\ -2 \end{pmatrix}$ and $\begin{pmatrix} 4 \\ 0 \\ -4 \end{pmatrix}$

2024 JPJC Vectors Tutorial Q12

2024 JPJC Vectors Tutorial Q12

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