Two straight lines, $l_1$ and $l_2$, have the equations
$l_1: \mathbf{r} = \mathbf{i} - 2\mathbf{j} + 3\mathbf{k} + \lambda(2\mathbf{i} - \mathbf{j} + a\mathbf{k})$
and
$l_2: \mathbf{r} = -\mathbf{i} - \mathbf{j} - \mathbf{k} + \mu(3\mathbf{i} - 2\mathbf{j} - 2\mathbf{k})$,
where $a$ is constant.
These lines, $l_1$ and $l_2$, are perpendicular.
(a)[1]
Show that the value of $a$ is $4$.
(b)[4]
Find the position vector of the point where the lines intersect.
(c)[2]
Point $A$ is given by the position vector $-5\mathbf{i} + \mathbf{j} - 9\mathbf{k}$. Show that $A$ lies on $l_1$.
(d)[2]
Point $B$ is the image of $A$ under reflection in the line $l_2$. Find the position vector of $B$.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Calculate the scalar product of the direction vectors, set it equal to zero, and obtain $a=4$” …