Two laser guidance beams used in a large-scale precision engineering project are modelled as lines l1 l_1\,l1 and l2 l_2\,l2 relative to a fixed origin OOO. The equations of the lines are given by:
l1:r=(i+kj)+t(i−2j+2k) l_1: \mathbf{r} = (\mathbf{i} + k\mathbf{j}) + t(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}) l1:r=(i+kj)+t(i−2j+2k) l2:r=(i+4j+6k)+s(2i+j−2k) l_2: \mathbf{r} = (\mathbf{i} + 4\mathbf{j} + 6\mathbf{k}) + s(2\mathbf{i} + \mathbf{j} - 2\mathbf{k}) l2:r=(i+4j+6k)+s(2i+j−2k)where t t\,t and s s\,s are scalar parameters and k k\,k is a constant.
Given that the two laser beams intersect at a single point:
find the value of kkk.
find the position vector of the point of intersection.
Calculate the acute angle between l1 l_1\,l1 and l2l_2l2. Give your answer in degrees to one decimal place.
The point A A\,A lies on l1 l_1\,l1 with parameter t=4t = 4t=4. The point B B\,B lies on l2 l_2\,l2 such that the vector AB⃗\vec{AB}AB is perpendicular to l2l_2l2.
Find the coordinates of BBB.
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.