3D math - vector

vector

Vector In physics, it is called vector , In mathematics, it is called vector .

Definition of vector ： Is an existing size And then there is Direction The amount of .

There are two ways to express , Just choose one , as follows
$$\begin{gathered} Row\; vector:\left[\begin{array}{ccc}{a}_1& a_2& a_3\end{array}\right] \\ Column\; vector:\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]\text{\hspace{1em}(1)} \end{gathered}$$

stay Unity The middle vector is Position（ Location ）, It can be either a point , Used to indicate position , It can also be a vector pointing to a certain position from the origin .

Tips ： Mathematically, point and vector are equivalent

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Zero vector
$$\vec{0}=\left[\begin{array}{cccc}0& 0& \cdots & 0\end{array}\right]$$
The zero vector is the only vector that has no direction , And the size is 0.

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Negative vector
$$-\vec{v}=-\left[\begin{array}{cccc}{a}_1& a_2& \cdots & a_n\end{array}\right]=\left[\begin{array}{cccc}-a_1& -a_2& \cdots & -a_n\end{array}\right]$$
Negative vectors are generated by equal but In the opposite direction Vector of

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Scalar And vector Of Relationship

Scalar is a number , They cannot be added directly to vectors , But it can be multiplied by a vector .(k For the scalar )
$$k\vec{v}=k\left[\begin{array}{cccc}{a}_1& a_2& \cdots & a_n\end{array}\right]=\left[\begin{array}{cccc}k{a}_1& k{a}_2& \cdots & k{a}_n\end{array}\right]$$
The result of multiplication is still a vector

• k > 0 when , Direction unchanged , The size changes to the original |k| times
• k = 0 when , Become a zero vector
• k < 0 when , In the opposite direction , The size changes to the original |k| times

Conclusion ：k > 0 Direction unchanged ,k < 0 In the opposite direction . The size is the original |k| times .

In actual development , We often Use scalar and unit vector multiplication , To obtain the vector of the corresponding size in a certain direction

Scalar division is similar to multiplication , I won't repeat

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Vector size

The size of the vector is also called the length of the vector , That is, calculate the distance from the origin of the vector to the target position .
$$\Vert \begin{array}{c}\vec{v}\end{array}\Vert =\sqrt{\sum _{i=1}^n{a}_i^2}=\sqrt{a_1^2+a_2^2+\cdots +a_n^2}$$

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Unit vector

The unit vector is of size 1 Vector of ,** They do not represent the size of the vector , It only represents the direction of the vector .** Also known as normalized vector .
$$\hat{v}=\frac{\vec{v}}{\Vert \begin{array}{c}\vec{v}\end{array}\Vert }$$

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Vector addition and Subtraction

Vectors can be added and subtracted .

Add ：
$$\vec{a}+\vec{b}=\left[\begin{array}{c}{a}_1+b_1\\ a_2+b_2\\ \cdots \\ a_n+b_n\end{array}\right]$$
The addition of vectors satisfies the commutative law . And the geometric meaning of vector addition is very important . They are not just the addition of values . what's more , Get a vector a With vector b The direction of extends the vector b Vector of magnitude . And the vector addition satisfies the triangle rule .

Addition is very common , They are often used to move vectors （ Such as how much distance to go in a certain direction ）

Subtraction ：
$$\vec{a}-\vec{b}=\left[\begin{array}{c}{a}_1-b_1\\ a_2-b_2\\ \cdots \\ a_n-b_n\end{array}\right]$$
Vector subtraction does not satisfy commutative law , in other words a - b and b - a The geometric meaning is different .

a - b The geometric meaning of is to get a b Point to a Vector of , And the vector size is a And b Distance of , The vector direction is b Point to a The direction of

contrary ,b - a The geometric meaning of is to get a a Point to b Vector of , And the vector size is a And b Distance of , The vector direction is a Point to b The direction of .

Subtraction is often used to find the distance between two positions , Or get the relative direction of two positions .

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Vector Dot product and Cross product

There are two kinds of multiplication between vectors , One is Dot product , One is Cross product . The most important difference between them is , The value of dot product is scalar , The value of the cross product is a vector .

Dot product ：
$$\begin{gathered} \vec{a}\cdot \vec{b}=\sum _{i=1}^n{a}_i{b}_i=a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n=\Vert \begin{array}{c}\vec{a}\end{array}\Vert \Vert \begin{array}{c}\vec{b}\end{array}\Vert \cos \theta \\ \theta For\; the\; angle\; between\; them \end{gathered}$$
The geometric meaning of dot product is very important , We will focus on

First of all Projection , As the most important characteristic of dot product , Through dot product, the projection size of one vector relative to another vector can be easily calculated .$\vec{a}\cdot \hat{b}$ be relative to $\vec{a}$ stay $\vec{b}$ The size of the projection on .
$$\vec{a}\cdot \hat{b}=\Vert \begin{array}{c}\vec{a}\end{array}\Vert \cos \theta$$
The projection gets scalar , That is, the size of the projection , If you want to get the vector of projection , You can multiply by the unit vector . as follows
$$\begin{gathered} k=\vec{a}\cdot \hat{b}=\Vert \begin{array}{c}\vec{a}\end{array}\Vert \cos \theta \\ \vec{v}=k\hat{b} \end{gathered}$$
What we get above is the vector a be relative to vector b Upper Horizontal vector . We can also find vectors a Relative to vector b Of Vertical vector
$$\begin{gathered} \vec{a}=a_{\parallel }+a_{\perp } \\ \text{\hspace{1em}(1)} \end{gathered}$$

$$a_{\parallel }=(\vec{a}\cdot \hat{b})\hat{b}\text{\hspace{1em}(2)}$$

$$\begin{array}{rl}{a}_{\perp }& =\vec{a}-a_{\parallel }\\ & =\vec{a}-(\vec{a}\cdot \hat{b})\hat{b}\end{array}\text{\hspace{1em}(3)}$$

This formula for horizontal and vertical projection is very useful .

Secondly, the dot product can also roughly judge the angle

• $\vec{a}\cdot \vec{b}$ > 0 It's an acute angle
• $\vec{a}\cdot \vec{b}$ = 0 Is a right angle
• $\vec{a}\cdot \vec{b}$ < 0 It's an obtuse angle

Another very important feature is that you can find the angle between two vectors
$$\theta =\arccos \frac{\vec{a}\cdot \vec{b}}{\Vert \begin{array}{c}\vec{a}\end{array}\Vert \Vert \begin{array}{c}\vec{b}\end{array}\Vert }$$
It can be simplified into
$$\theta =\arccos (\hat{a}\cdot \hat{b})$$
Cross product （ The three dimensional ）：
$$\vec{a}\times \vec{b}=\left[\begin{array}{ccc}i& x_1& x_2\\ j& y_1& y_2\\ k& z_1& z_2\end{array}\right]=\left[\begin{array}{c}(y_1{z}_2-y_2{z}_1)i\\ (z_1{x}_2-z_2{x}_1)j\\ (x_1{y}_2-x_2{y}_1)k\end{array}\right]=\left[\begin{array}{c}{y}_1{z}_2-y_2{z}_1\\ z_1{x}_2-z_2{x}_1\\ x_1{y}_2-x_2{y}_1\end{array}\right]$$

$$i=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]j=\left[\begin{array}{ccc}0& 1& 0\end{array}\right]k=\left[\begin{array}{ccc}0& 0& 1\end{array}\right]$$

The result of cross product is vector , The obtained vector is ** The normal vector of the plane composed of two vectors .** This is also the most important use of cross product , Get the normal vector of the plane composed of two vectors .

Other formulas of cross product
$$\Vert \begin{array}{c}\vec{a}\times \vec{b}\end{array}\Vert =\Vert \begin{array}{c}\vec{a}\end{array}\Vert \Vert \begin{array}{c}\vec{b}\end{array}\Vert \sin \theta$$
This formula calculates the area of a quadrilateral composed of two vectors . But it's less used .

The cross product can use the right-hand rule to determine the direction of the normal vector .

Empathy , You can judge whether the two vectors are clockwise by the normal vector , Or counter clockwise