Read a text vector product inside and outside - Code World

Read a text vector product inside and outside

Others 2020-01-11 23:47:16 views: null

table of Contents

內积（点积） Dot Product; Scalar Product;inner Product

Outer product (cross product) Exterior Product

Tensor product Outer Product

Foreword

There are three-dimensional vector $\large \boldsymbol{a}=(a_1,a_2,a_3)$ $\large \boldsymbol{b}=(b_1,b_2,b_3)$

tips: The output of the input three kinds of product can be resolved.

內积（点积） Dot Product; Scalar Product;inner Product

$\large \boldsymbol{a\cdot b}=\boldsymbol{a^T b}=\begin{bmatrix} a_1&a_2&a_3\end{bmatrix}\begin{bmatrix} b_1\\b_2\\b_3\end{bmatrix}=\sum_{i=1}^{3}{a_ib_i}=\left | \boldsymbol{a} \right | \left | \boldsymbol{b}\right | cos \left \langle \boldsymbol{a,b} \right \rangle$

where, $\large \left | \boldsymbol{a} \right |$ represents a number of the vector 2-norm (length), $\large cos \left \langle \boldsymbol{a,b} \right \rangle$ it represents an angle.

Representation between the vector projection relationship.

Note: inner Product two input vectors, scalar outputs

Outer product (cross product) Exterior Product

The concept of Analytic Geometry Analytic Geometry

$\large \boldsymbol{a\times b}=\begin{Vmatrix} i & j& k\\ a_1&a_2 & a_3\\ b_1& b_2 & b_3 \end{Vmatrix}=\begin{bmatrix} a_2b_3-a_3b_2\\ a_3b_1-a_1b_3 \\ a_1b_2-a_2b_1 \end{bmatrix}$

among them, $\large i=(1{,0,0})$ $\large j=(0{,1,0})$ $\large k=(0{,0,1})$

which is $\large \boldsymbol{a\times b}=(a_2b_3-a_3b_2 )i-(a_3b_1-a_1b_3 )j+(a_1b_2-a_2b_1) k$

Note: two cross product of the input vector, output vector a (instead of a scalar), and and $\large \boldsymbol{a b}$ perpendicular to the plane formed.

Tensor product Outer Product

Modern concepts in linear algebra,

$\large \boldsymbol{a\cdot b}=\boldsymbol{a b^T}=\begin{bmatrix} a_1\\a_2\\a_3\end{bmatrix}\begin{bmatrix} b_1&b_2&b_3\end{bmatrix}=\begin{bmatrix} a_1b_1 &a_1b_2 &a_1b_3 \\ a_2b_1 &a_2b_2 &a_2b_3 \\ a_3b_1 &a_3b_2 &a_3b_3 \end{bmatrix}$

Note: Outer Product two input vectors, outputs a matrix

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