Definition:Dot Product

Definition

Given any two vectors $a$ and $b$ in $\R^n$, the Dot Product is defined as:

$\displaystyle \mathbf a \cdot \mathbf b = a_1 b_1 + a_2 b_2 + \cdots + a_n b_n = \sum_{i=1}^{n} a_i b_i$

If the vectors are represented as column matrices:

$\mathbf a = \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{bmatrix} , \mathbf b = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix}$

we can express the dot product as:

$\mathbf a \cdot \mathbf b = \mathbf a^T \mathbf b$

where:

• $\mathbf a^T = \begin{bmatrix} a_1 & a_2 & \cdots & a_n \end{bmatrix}$ is the transpose of $\mathbf a$
• the operation between the matrices is the matrix product.

Alternative Definition

Given any two vectors $a$ and $b$ in $\R^n$, the Dot Product can be defined as:

$\mathbf a \cdot \mathbf b = \left\Vert{ \mathbf a }\right\Vert \left\Vert{ \mathbf b }\right\Vert \cos \angle \mathbf a , \mathbf b$

where:

• $\left\Vert{ \mathbf a }\right\Vert$ is the length of $\mathbf a$ and $\left\Vert{ \mathbf b }\right\Vert$ is the length of $\mathbf b$
• $\angle \mathbf a, \mathbf b$ is the angle between $\mathbf a$ and $\mathbf b$, taken to be between $0$ and $\pi$.

It can be shown that these two definitions are equivalent.

Also known as

The dot product is also known as:

• The Scalar Product (but this can be confused with multiplication by a scalar so is less recommended)
• The Standard Inner Product.

The symbol used for the dot is variously presented; another version is $\mathbf a \bullet \mathbf b$, which can be preferred if there is ambiguity between the dot product and standard multiplication.

Also see

• Properties of Dot Product
• Dot Product is Inner Product