Inner Product of Vectors

The inner product of two vectors is a binary operation which takes two vectors and produces a scalar.

Let $\vec{u},\vec{v}\in V={\mathbb{R}}^n$, then the inner product of $\vec{u}$ and $\vec{v}$, denoted $\vec{u}\cdot \vec{v}$ or $⟨\vec{u},\vec{v}⟩$, is defined as:

$$\vec{u}\cdot \vec{v}=u_1{v}_1+u_2{v}_2+\cdots +u_n{v}_n=\sum _{i=1}^n{u}_i{v}_i$$

where $\vec{u}=(u_1,u_2,\dots ,u_n)$ and $\vec{v}=(v_1,v_2,\dots ,v_n)$.

Computing the Angle between Vectors

The inner product can be used to compute the angle $\theta$ between two vectors $\vec{u}$ and $\vec{v}$ using the formula:

$$\vec{u}\cdot \vec{v}=\Vert \vec{u}\Vert \Vert \vec{v}\Vert \cos (\theta )$$

where $\Vert \vec{u}\Vert$ and $\Vert \vec{v}\Vert$ are the magnitude of the vectors$\vec{u}$ and $\vec{v}$ respectively.

Rearranging the formula gives:

$$\theta =\arccos \left(\frac{\vec{u}\cdot \vec{v}}{\Vert \vec{u}\Vert \Vert \vec{v}\Vert }\right)$$

Properties

1. Commutative Property: $\vec{u}\cdot \vec{v}=\vec{v}\cdot \vec{u}$
2. Distributive Property: $\vec{u}\cdot (\vec{v}+\vec{w})=\vec{u}\cdot \vec{v}+\vec{u}\cdot \vec{w}$
3. Scalar Multiplication: $(c\vec{u})\cdot \vec{v}=\vec{u}\cdot (c\vec{v})=c(\vec{u}\cdot \vec{v})$
4. Positive Definiteness: $\vec{u}\cdot \vec{u}\ge 0$ with equality if and only if $\vec{u}=\vec{0}$
5. If $\vec{u}\cdot \vec{u}=0$ then $\vec{u}=\vec{0}$