With respect to a fixed origin OOO, the points AAA and BBB have position vectors
(231) and (4−14) \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} \text{ and } \begin{pmatrix} 4 \\ -1 \\ 4 \end{pmatrix} 231 and 4−14respectively. The line l1l_1l1 passes through the points AAA and BBB.
Write down an equation for l1l_1l1 in the form r=p+λq\mathbf{r} = \mathbf{p} + \lambda \mathbf{q}r=p+λq.
The line l2l_2l2 has equation
r=(12−3)+μ(1−12) \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} r=12−3+μ1−12Show that l1l_1l1 and l2l_2l2 do not meet.
The point CCC is on l2l_2l2 where μ=2\mu = 2μ=2.
Find the acute angle between the line segment ACACAC and the line l2l_2l2. Give your answer in degrees to one decimal place.
197 exam-style questions on OCR A Level Maths 1.10 Vectors, covering 1.10.1 Vectors in two dimensions, 1.10.2 Vectors in three dimensions (A-level only), 1.10.3 Magnitude and direction of vectors, 1.10.4 Basic operations on vectors, 1.10.5 Position vectors, 1.10.6 Distance between points, 1.10.7 Problem solving using vectors, 1.10.8 Vectors in kinematics, and 1.10 Vectors. Each one has a worked solution and a mark scheme showing where the marks go.