The two lines are described by $l_1:\ \mathbf{r}=(2\mathbf{i}+\mathbf{j}+4\mathbf{k})+\lambda(\mathbf{i}+2\mathbf{j}-3\mathbf{k})$, $l_2:\ \mathbf{r}=(3\mathbf{i}-\mathbf{j}+5\mathbf{k})+\mu(2\mathbf{i}+3\mathbf{j}+a\mathbf{k})$.
(a)[2]
Find the value of $a$ such that $l_1$ is perpendicular to $l_2$.
(b)[4]
Find the value of $a$ such that $l_1$ and $l_2$ intersect.
(c)[4]
Find the values of $a$ for which the acute angle between $l_1$ and $l_2$ equals $\cos^{-1}\!\left(\frac{5}{14}\right)$.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Calculate the scalar product of the direction vectors to obtain an equation in $a$.” …