9. Dot Product

The dot product (or scalar product) of two vectors of the same dimension returns a scalar — the sum of componentwise products.

9.1. Geometric formula

where is the angle between and — and is the norm.

This links the algebraic dot product to geometric intuition: the dot product measures how much the vectors point in the same direction.

9.2. Connection to norm

The square of the norm is the dot product of a vector with itself:

See Norm for the full story.

9.3. Properties

Commutative:

Distributive over vector addition:

Compatible with scalar multiplication:

Bilinear: linear in each argument separately (the three properties above combined).

9.4. Matrix form

For column vectors, the dot product is just matrix multiplication of with :

(See Transpose.)

9.5. See also

• Norm —
• Angles Between Vectors —
• Cauchy–Schwarz Inequality
• Triangle Inequality
• Orthogonality —
• Cross Product + Dot vs Cross
• Inner Product — generalization