Dot Product

The dot product (inner product) of two vectors:

$$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n$$

Geometric Meaning

$$\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos\theta$$

Where $\theta$ is the angle between the vectors.

Key Results

• Perpendicular: $\mathbf{u} \cdot \mathbf{v} = 0$ means the vectors are orthogonal
• Same direction: $\mathbf{u} \cdot \mathbf{v} > 0$
• Opposite direction: $\mathbf{u} \cdot \mathbf{v} < 0$
• Length: $\mathbf{v} \cdot \mathbf{v} = \|\mathbf{v}\|^2$

Properties

• Commutative: $\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}$
• Distributive: $\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}$
• Bilinear: $(c\mathbf{u}) \cdot \mathbf{v} = c(\mathbf{u} \cdot \mathbf{v})$