Two research submersibles, Alpha and Beta, are mapping a deep-sea trench. Their paths are modeled as straight lines relative to a sonar station at the origin OOO. The position vectors of the submersibles, in meters, are given by
rA=(i−j+5k)+λ(2i+j+3k)\mathbf{r}_A = (\mathbf{i} - \mathbf{j} + 5\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} + 3\mathbf{k})rA=(i−j+5k)+λ(2i+j+3k) rB=(4i+3j)+μ(i−2j+pk)\mathbf{r}_B = (4\mathbf{i} + 3\mathbf{j}) + \mu(\mathbf{i} - 2\mathbf{j} + p\mathbf{k})rB=(4i+3j)+μ(i−2j+pk)
where λ\lambdaλ and μ\muμ are scalar parameters and ppp is a constant.
Show that for all values of p≠11p \neq 11p=11, the paths of the submersibles are skew.
Practise Edexcel A Level Maths Vectors with exam-style questions for A Level Maths. 100 questions covering 12.1 3D Coordinates, 12.2 Vectors in 3D, 12.3 Solving Geometric Problems, and 12.4 Application to Mechanics, matched to the Edexcel A Level Maths (9MA0) 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.