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.
The point D D\,D is such that AB⃗=BD⃗\vec{AB} = \vec{BD}AB=BD.
Find the position vector of DDD.
Given that ∣AC⃗∣=5\left|\vec{AC}\right| = 5AC=5, find the exact value of aaa.
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.