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