The line $l$ is given by $\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(a\mathbf{i} + 2\mathbf{j} + \mathbf{k})$, with $a$ constant. The plane $p$ is defined by $x + 2y + 2z = 6$. Determine the value or values of $a$ for each of the situations below.
(a(i))[2]
The line $l$ runs parallel to the plane $p$.
(a(ii))[4]
The line $l$ cuts the line that passes through the points whose position vectors are $3\mathbf{i} + 2\mathbf{j} + \mathbf{k}$ and $\mathbf{i} + \mathbf{j} - \mathbf{k}$.
(a(iii))[5]
The acute angle between the line $l$ and the plane $p$ is $\tan^{-1} 2$.
Worked solution & mark scheme
This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Set the scalar product of the direction vectors of $l$ and $p$ equal to zero” …