Definition:Vector Product

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Definition

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.


Cross Product

Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $3$ dimensions:

\(\ds \mathbf a\) \(=\) \(\ds a_i \mathbf i + a_j \mathbf j + a_k \mathbf k\)
\(\ds \mathbf b\) \(=\) \(\ds b_i \mathbf i + b_j \mathbf j + b_k \mathbf k\)

where $\tuple {\mathbf i, \mathbf j, \mathbf k}$ is the standard ordered basis of $\mathbf V$.


The vector cross product, denoted $\mathbf a \times \mathbf b$, is defined as:

$\mathbf a \times \mathbf b = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k\\ a_i & a_j & a_k \\ b_i & b_j & b_k \\ \end{vmatrix}$

where $\begin {vmatrix} \ldots \end {vmatrix}$ is interpreted as a determinant.

More directly:

$\mathbf a \times \mathbf b = \paren {a_j b_k - a_k b_j} \mathbf i - \paren {a_i b_k - a_k b_i} \mathbf j + \paren {a_i b_j - a_j b_i} \mathbf k$


Outer Product

Given two vectors $\mathbf u = \tuple {u_1, u_2, \ldots, u_m}$ and $\mathbf v = \tuple {v_1, v_2, \ldots, v_n}$, their outer product $\mathbf u \otimes \mathbf v$ is defined as:

$\mathbf u \otimes \mathbf v = A = \begin{bmatrix} u_1 v_1 & u_1 v_2 & \dots & u_1 v_n \\ u_2 v_1 & u_2 v_2 & \dots & u_2 v_n \\ \vdots & \vdots & \ddots & \vdots \\ u_m v_1 & u_m v_2 & \dots & u_m v_n \end{bmatrix}$


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