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+15j+νk\mathbf{c} = \mu\mathbf{i} + 15\mathbf{j} + \nu\mathbf{k}c=μi+15j+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.
168 exam-style questions on AQA A Level Maths 1.13 J: Vectors, covering 1.13.1 Vectors in two and three dimensions, 1.13.2 Magnitude and direction of a vector, 1.13.3 Vector arithmetic, 1.13.4 Position vectors, and 1.13.5 Vectors to solve problems. Each one has a worked solution and a mark scheme showing where the marks go.