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.
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.