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.
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.