Dot Product

The dot product (inner product) of two vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R}^n$ can be understood through two equivalent views:

1. Algebraic view — multiply corresponding entries and sum: $\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i$
2. Geometric view — project one vector onto the other: $\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta$, where $\theta$ is the angle between them

The dot product is commutative $(\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a})$, distributive over addition, and linear in each argument.

Key geometric facts:

• $\mathbf{a} \cdot \mathbf{b} > 0$ means the vectors point in roughly the same direction
• $\mathbf{a} \cdot \mathbf{b} = 0$ means the vectors are orthogonal (perpendicular)
• $\mathbf{a} \cdot \mathbf{b} < 0$ means they point in roughly opposite directions
• $\mathbf{a} \cdot \mathbf{a} = \|\mathbf{a}\|^2$ gives the squared length of the vector

The dot product is the building block of matrix multiplication: each entry of $AB$ is a dot product of a row of $A$ with a column of $B$.

See also: Matrix Multiplication, Rank and Null Space, Eigendecomposition