vector

Object that has magnitude (or length) and direction.
Equivalent: Euclidian vector, spatial vector, geometric vector

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Length / magnitude / euclidean norm of vector $a$

$$||(x_1,x_3,...,x_n)||=\sqrt{\sum _{i=1}^n{x}_i^2}$$

Unit vector

Vector with length of 1.
Indicates direction.
On the unit sphere.
Normalize a vector / divide vector by its length to get the unit vector:

\hat{a} = \frac{a}{||a||} $$ ^cf975b

High dimensional vectors

The alignment / difference in angle / direction of vectors is much more descriptive than the magnitude of a vector.
Take a 1k dimensional vector, it has 999 degrees of freedom to rotate and just 1 direction for length.

Cauchy-Schwarz inequality

$$|a\cdot b|\le ||a||\cdot ||b||$$

$|\cos (\theta )|\le 1⟹|{\mathbf{a}}^T\mathbf{b}|=\Vert \mathbf{a}\Vert \cdot \Vert \mathbf{b}\Vert \cos (\theta )\le \Vert \mathbf{a}\Vert \cdot \Vert \mathbf{b}\Vert$