Dot Product

The dot product, also called scalar product, and rarely, the Euclidean inner product is an operation that can be performed on Vectors to return a scalar/real number. It can also be used to check if vectors are parallel or perpendicular.

The dot product can only be applied if both vectors are tuples of the same size.

Definition §

Formula §

$$\vec{u}\cdot \vec{v}=\sum _{k=1}^n{u}_k{v}_k=u_1{v}_1+u_2{v}_2+\dots u_n{v}_n$$

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

• $\vec{u}$, $\vec{v}$ = Two vectors $n$ dimensions (i.e. they belong to ${\mathbb{R}}^n$):
• $\theta$ = The angle between $\vec{u}$ and $\vec{v}$

Matrix Formula §

$$\vec{u}\cdot \vec{v}={\mathbf{U}}^T\mathbf{V}$$

• $\mathbf{U},\mathbf{V}$ = The associated column matrices of $\vec{u}$ and $\vec{v}$

Properties §

Let $\vec{u},\vec{v},\vec{w}\in {\mathbb{R}}^n$ and let $k\in \mathbb{R}$ (a scalar). Then:

1. The dot product of two vectors returns a scalar:$$\vec{u} \cdot \vec{v} \in \mathbb{R}

2. **Commutative**: The order of vectors in a dot product can be reversed: $$\vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u}

3. Distributive: Dot products can be expanded:$$(\vec{u} + \vec{v})\cdot \vec{w} = \vec{u}\cdot \vec{w} + \vec{v}\cdot \vec{w}

$$4.\ast \ast Non-Associative!\ast \ast :Weca{n}^{\prime }trearrangetheparenthesisindotproductoperations:$$

(\vec{u} + \vec{v})\cdot \vec{w} \neq \vec{u}\cdot(\vec{v} + \vec{w}) \ \ \ \ \ \ \ \ \ \ \ \vec{u}\cdot(\vec{v} + \vec{w}) = \vec{u}\cdot \vec{v} + \vec{u}\cdot \vec{w}

6. **Obeys scalar multiplication**: We can 'pull out' the scalar from a dot product operation:$$(k\vec{u})\cdot \vec{v} = \vec{u}\cdot(k\vec{v})= k(\vec{u} \cdot \vec{v})

7. The dot product of a vector with itself returns it’s magnitude squared:$$\vec{u}\cdot \vec{u} = \lvert\lvert \vec{u} \rvert\rvert

8. **Orthogonality**: The dot product of two vectors is zero if they are [perpendicular](Vector.md#Perpendicular) or orthogonal:$$\vec{u}\cdot \vec{v} = 0 \iff \vec{u} \perp \vec{v}

Geometry §

#todo

Cauchy-Schwarz inequality §

Correlation §

#todo