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

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

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

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