Mathematics 9709 · AS & A Level · Coordinate geometry
Coordinate geometry — practice question
With origin $O$, the position vectors of $A$, $B$ and $C$ are $\vec{OA} = \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$, $\vec{OB} = \begin{pmatrix} 5 \\ -1 \\ k \end{pmatrix}$ and $\vec{OC} = \begin{pmatrix} 2 \\ 6 \\ -3 \end{pmatrix}$ respectively, where $k$ is a constant.
(i)[2]
Determine the value of $k$ when $\angle AOB = 90^\circ$.
(ii)[4]
Find the possible values of $k$ for which the lengths of $AB$ and $OC$ are equal.
(iii)[4]
Point $D$ lies in the same direction as $\vec{OA}$, with $\vec{OD}$ having magnitude $9$ units. Point $E$ lies in the same direction as $\vec{OC}$, with $\vec{OE}$ having magnitude $14$ units. Find the magnitude of $\vec{DE}$ in the form $\sqrt{n}$ where $n$ is an integer.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Apply the scalar product condition $\vec{OA}\cdot(\vec{OB}-\vec{OC})=0$.” …