Algebra and Geometry | Part IA, 2001

Show, using the summation convention or otherwise, that a×(b×c)=(ac)b\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}- (a.b)c, for a, b, c R3\in \mathbb{R}^{3}

The function Π:R3R3\Pi: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} is defined by Π(x)=n×(x×n)\Pi(\mathbf{x})=\mathbf{n} \times(\mathbf{x} \times \mathbf{n}) where n\mathbf{n} is a unit vector in R3\mathbb{R}^{3}. Show that Π\Pi is linear and find the elements of a matrix PP such that Π(x)=Px\Pi(\mathbf{x})=P \mathbf{x} for all xR3\mathbf{x} \in \mathbb{R}^{3}.

Find all solutions to the equation Π(x)=x\Pi(\mathbf{x})=\mathbf{x}. Evaluate Π(n)\Pi(\mathbf{n}). Describe the function П geometrically. Justify your answer.

Typos? Please submit corrections to this page on GitHub.