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.
168 exam-style questions on OCR (MEI) A Level Maths 1.11 Vectors, covering 1.11.1 Language of vectors in two dimensions, 1.11.2 Add, subtract and scale vectors, 1.11.3 Magnitude and direction of a vector, 1.11.4 Position vectors, 1.11.5 Distance between points by position vectors, 1.11.6 Vectors to solve problems, 1.11.7 Language of vectors in three dimensions (A-level only), and 1.11 Vectors. Each one has a worked solution and a mark scheme showing where the marks go.