A maritime surveillance drone tracks two suspicious vessels. Relative to a coastal monitoring station at the origin OOO, the first vessel's position is recorded at point A(3,−2,1)A(3, -2, 1)A(3,−2,1) and subsequently at point B(5,2,3)B(5, 2, 3)B(5,2,3), where coordinates are measured in kilometres. It is assumed the vessel travels along a straight line l1l_1l1 passing through 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 second vessel follows a path l2l_2l2 defined by the equation
r=(014)+μ(21−1) \mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix} r=014+μ21−1Show that the paths of the two vessels, l1l_1l1 and l2l_2l2, do not meet.
The monitoring station identifies a specific point CCC on the path l2l_2l2 where the parameter μ=−1\mu = -1μ=−1.
Find the acute angle between the line segment ACACAC and the path l2l_2l2. Give your answer in degrees to one decimal place.
Practise AQA A Level Maths 1.13 J: Vectors with exam-style questions for A Level Maths. 122 questions covering 1.13.1 Vectors in two and three dimensions, 1.13.2 Magnitude and direction of a vector, 1.13.3 Vector arithmetic, 1.13.4 Position vectors, and 1.13.5 Vectors to solve problems, matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
