# Category:Dot Product

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This category contains results about **the vector dot product**.

Definitions specific to this category can be found in Definitions/Dot Product.

Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $n$ dimensions:

\(\ds \mathbf a\) | \(=\) | \(\ds \sum_{k \mathop = 1}^n a_k \mathbf e_k\) | ||||||||||||

\(\ds \mathbf b\) | \(=\) | \(\ds \sum_{k \mathop = 1}^n b_k \mathbf e_k\) |

where $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is the standard ordered basis of $\mathbf V$.

The **dot product** of $\mathbf a$ and $\mathbf b$ is defined as:

\(\ds \mathbf a \cdot \mathbf b\) | \(:=\) | \(\ds \sum_{k \mathop = 1}^n a_k b_k\) | ||||||||||||

\(\ds \mathbf b\) | \(=\) | \(\ds a_1 b_1 + a_2 b_2 + \cdots + a_n b_n = \sum_{i \mathop = 1}^n a_i b_i\) |

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:

- $\mathbf a \cdot \mathbf b = \mathbf a^\intercal \mathbf b$

where:

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

## Subcategories

This category has the following 15 subcategories, out of 15 total.

## Pages in category "Dot Product"

The following 32 pages are in this category, out of 32 total.

### D

- Derivative of Dot Product of Vector-Valued Functions
- Derivative of Square of Vector-Valued Function
- Dot Product Associates with Scalar Multiplication
- Dot Product Distributes over Addition
- Dot Product is Inner Product
- Dot Product of Constant Magnitude Vector-Valued Function with its Derivative is Zero
- Dot Product of Elements of Standard Ordered Basis
- Dot Product of Like Vectors
- Dot Product of Orthonormal Basis Vectors
- Dot Product of Perpendicular Vectors
- Dot Product of Sum with Difference of Vectors
- Dot Product of Unit Vectors
- Dot Product of Vector Cross Products
- Dot Product of Vector with Itself
- Dot Product of Vector-Valued Function with its Derivative
- Dot Product Operator is Bilinear
- Dot Product Operator is Commutative
- Dot Product with Self is Non-Negative
- Dot Product with Self is Zero iff Zero Vector
- Dot Product with Zero Vector is Zero