Points $A$, $B$ and $C$ are given by position vectors $\overrightarrow{OA} = -2\mathbf{i} + \mathbf{j} + 4\mathbf{k}$, $\overrightarrow{OB} = 5\mathbf{i} + 2\mathbf{j}$ and $\overrightarrow{OC} = 8\mathbf{i} + 5\mathbf{j} - 3\mathbf{k}$, with $O$ as the origin. The line $l_1$ goes through $B$ and $C$.
(a)[3]
Find a vector equation of $l_1$.
(b)[4]
The line $l_2$ is given by $\mathbf{r} = -2\mathbf{i} + \mathbf{j} + 4\mathbf{k} + \mu(3\mathbf{i} + \mathbf{j} - 2\mathbf{k})$. Determine the coordinates of the intersection of $l_1$ and $l_2$.
(c)[5]
Point $D$ lies on $l_2$ with $AB = BD$. Determine the position vector of $D$.
Worked solution & mark scheme
This 12-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Give a correct direction vector for $\overrightarrow{BC}$, e.g. $3\mathbf{i}+3\mathbf{j}-3\mathbf{k}$” …