7、Understanding Vector Geometry: Lengths, Angles, and Projections

Understanding Vector Geometry: Lengths, Angles, and Projections

1. Lengths and Distances

Inner products and norms are closely related. Any inner product can naturally induce a norm, as shown by the formula $|x| := \sqrt{\langle x, x \rangle}$. However, not all norms are induced by inner products; the Manhattan norm is an example.

1.1 Cauchy - Schwarz Inequality

For an inner product vector space $(V, \langle \cdot, \cdot \rangle)$, the induced norm $|\cdot|$ satisfies the Cauchy - Schwarz inequality: $|\langle x, y \rangle| \leq |x| |y|$.

1.2 Example of Vector Lengths

Let’s take the vector $x = [1, 1]^\top \in \mathbb{R}^2$.
- Using the dot product as the inner product, $|x| = \sqrt{x^\top x} = \sqrt{1^2 + 1^2} = \