Paper 1, Section II, 5V5 \mathbf{V}

Vectors and Matrices | Part IA, 2018

Let x,yRn\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n} be non-zero real vectors. Define the inner product xy\mathbf{x} \cdot \mathbf{y} in terms of the components xix_{i} and yiy_{i}, and define the norm x|\mathbf{x}|. Prove that xyxy\mathbf{x} \cdot \mathbf{y} \leqslant|\mathbf{x}||\mathbf{y}|. When does equality hold? Express the angle between x\mathbf{x} and y\mathbf{y} in terms of their inner product.

Use suffix notation to expand (a×b)(b×c)(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{b} \times \mathbf{c}).

Let a,b,c\mathbf{a}, \mathbf{b}, \mathbf{c} be given unit vectors in R3\mathbb{R}^{3}, and let m=(a×b)+(b×c)+(c×a)\mathbf{m}=(\mathbf{a} \times \mathbf{b})+(\mathbf{b} \times \mathbf{c})+(\mathbf{c} \times \mathbf{a}). Obtain expressions for the angle between m\mathbf{m} and each of a,b\mathbf{a}, \mathbf{b} and c\mathbf{c}, in terms of a,b,c\mathbf{a}, \mathbf{b}, \mathbf{c} and m|\mathbf{m}|. Calculate m|\mathbf{m}| for the particular case when the angles between a,b\mathbf{a}, \mathbf{b} and c\mathbf{c} are all equal to θ\theta, and check your result for an example with θ=0\theta=0 and an example with θ=π/2\theta=\pi / 2.

Consider three planes in R3\mathbb{R}^{3} passing through the points p,q\mathbf{p}, \mathbf{q} and r\mathbf{r}, respectively, with unit normals a,b\mathbf{a}, \mathbf{b} and c\mathbf{c}, respectively. State a condition that must be satisfied for the three planes to intersect at a single point, and find the intersection point.

Typos? Please submit corrections to this page on GitHub.