The points AAA, B B\,B and C C\,C have position vectors
a=2i+3j−k\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k}a=2i+3j−k, b=5i−j+2k\mathbf{b} = 5\mathbf{i} - \mathbf{j} + 2\mathbf{k}b=5i−j+2k, c=μi+11j+νk\mathbf{c} = \mu\mathbf{i} + 11\mathbf{j} + \nu\mathbf{k}c=μi+11j+νk
where μ \mu\,μ and ν \nu\,ν are constants.
Find AB⃗\vec{AB}AB and ∣AB⃗∣\left|\vec{AB}\right|AB, giving the magnitude in exact form.
Given that AC⃗\vec{AC}AC is parallel to AB⃗\vec{AB}AB, find the value of μ \mu\,μ and the value of ν\nuν.
Hence write down the ratio AB:ACAB : ACAB:AC.
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.