Dot product

The dot product (also the scalar or inner product) is a map on a vector space $\Sigma$ to a scalar $⟨\cdot ,\cdot ⟩:\Sigma \times \Sigma \to \Lambda$. Vectors are orthogonal if their dot product is zero.

Our most broadly applicable formula is:

$$\vec{v}\cdot \vec{w}=\sum _{i=1}^n{v}_i{w}_i=v_1{w}_1+v_2{w}_2+\cdots$$

We can also relate the angle between two vectors as follows:

$$\vec{v}\cdot \vec{w}=\cos (x)\Vert \vec{v}\Vert \Vert \vec{w}\Vert$$

We use in NumPy with np.dot(a, b) and in MATLAB with dot(a, b).

Properties

$$\vec{v}\cdot \vec{v}=\Vert \vec{v}{\Vert }^2$$

The dot product is commutative and follows conjugate symmetry, i.e.,

$$\vec{v}\cdot \vec{w}=\vec{w}\cdot \vec{v}$$

Extensions

Some convenient inequalities:

• Cauchy-Schwarz inequality

$$|v\cdot w|\le \Vert v\Vert \Vert w\Vert$$

• Triangle inequality

$$\Vert v+w\Vert \le \Vert v\Vert +\Vert w\Vert$$

See also

• Cross product