Relative to a fixed origin OOO, the point A A\,A has position vector (3i+4j−2k)(3\mathbf{i}+4\mathbf{j}-2\mathbf{k})(3i+4j−2k), the point B B\,B has position vector (7i−5j+3k)(7\mathbf{i}-5\mathbf{j}+3\mathbf{k})(7i−5j+3k), and the point C C\,C has position vector (ai+3j−4k)(a\mathbf{i}+3\mathbf{j}-4\mathbf{k})(ai+3j−4k), where a a\,a is a constant and a>0a>0a>0. D D\,D is the point such that AB⃗=BD⃗\vec{AB}=\vec{BD}AB=BD
Find AB⃗\vec{AB}AB.
Hence find the position vector of DDD.
Given that ∣AC⃗∣=5|\vec{AC}|=5∣AC∣=5, find the value of aaa.
309 exam-style questions on Edexcel A Level Maths Vectors, covering 12.1 3D Coordinates, 12.2 Vectors in 3D, 12.3 Solving Geometric Problems, and 12.4 Application to Mechanics. Each one has a worked solution and a mark scheme showing where the marks go.