vector

# Definitions and related concepts

vector: A quantity that has both magnitude and direction is called a vector. The vector studied mathematically is free vector, that is, as long as its size and direction are not changed, the starting point and end point can be moved in parallel at will. recorded as $\vec a$ or $\boldsymbol{a}$。

directed line segment: A line segment with a direction is called a directed line segment. A directed line segment has three elements:starting point, direction, length, knowing the three elements, the end point is uniquely determined. We represent vectors as directed line segments.

vector module: Directed line segment $\overrightarrow{AB}$ The length of is called the module of the vector, which is the size of this vector. Recorded as:$|\overrightarrow{AB}|$ or $|\boldsymbol{a}|$。

zero vector:Module is $0$ vector. The direction of the zero vector is arbitrary. Recorded as:$\vec 0$ or $\boldsymbol{0}$。

unit vector:Module is $1$ The vector is called the unit vector in that direction.

parallel vector: Two with the same or opposite directions non-zero vector. Recorded as:$\boldsymbol a\parallel \boldsymbol b$. For multiple vectors that are parallel to each other, a straight line can be drawn parallel to these vectors. Then any set of parallel vectors can be translated to the same straight line, so parallel vectors are also called collinear vector。

equal vectors: Vectors of equal magnitude and direction.

opposite vector: Vectors with equal modules and opposite directions.

angle between vectors: Two known non-zero vectors $\boldsymbol a,\boldsymbol b$,do $\overrightarrow{OA}=\boldsymbol a,\overrightarrow{OB}=\boldsymbol b$,So $\theta=\angle AOB$ It's a vector $\boldsymbol a$ with vector $\boldsymbol b$ angle. Recorded as:$\langle \boldsymbol a,\boldsymbol b\rangle$. Obviously when $\theta=0$ When the two vectors are in the same direction,$\theta=\pi$ When the two vectors are in opposite directions,$\theta=\frac{\pi}{2}$ When we say that two vectors are perpendicular, write them as $\boldsymbol a\perp \boldsymbol b$. And, we stipulate $\theta \in [0,\pi]$。

Note that plane vectors are directional, and we cannot compare the magnitude of two vectors (but we can compare the modulus lengths of two vectors). But two vectors can be equal.

# Linear operations on vectors

# Addition and subtraction of vectors

When we define a quantity, we want it to have operations. The operations of vectors can be compared to the operations of numbers, but we study the operations of vectors from the perspective of physics.

Analogy to the concept of displacement in physics, if a person starts from $A$ by $B$ Go to $C$, we say that the displacement he passed through is $\overrightarrow{AB}+\overrightarrow{BC}$, which is actually equivalent to the person directly getting the $A$ Go to $C$,Right now $\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$。

Note that the law of force synthesis, the parallelogram law, can also be viewed as the addition of some vectors.

So let’s sort out the addition rules for vectors:

1. Triangle Rule for Vector Addition: If the vectors required to be summed are connected end to end, then the sum of these vectors points from the starting point of the first vector to the end point of the last vector;
2. Parallelogram Rule for Vector Addition: If you want to sum two vectors common starting point, then their sum vector is the diagonal of the parallelogram with these two vectors as adjacent sides, the starting point is the common starting point of the two vectors, and the direction is along the diagonal direction of the parallelogram.

In this way, the addition of vectors has geometric meaning. And it can be verified that the addition of vectors satisfies commutative and associative laws。

Since subtracting real numbers can be written as adding their opposites, we consider subtracting vectors the same way. Right now:$\boldsymbol a-\boldsymbol b=\boldsymbol a+(-\boldsymbol b)$。

In this way, we consider the vectors with the same starting point and make their difference according to the parallelogram rule. After translation, we can find The "difference vector of the common starting point vector" is the directed line segment from the "subtracted vector" to the "subtracted vector"。

This is also the geometric meaning of vector subtraction.

We sometimes have two points $A,B$, want to know $\overrightarrow{AB}$, you can use the subtraction operation $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$ get.

# Multiplication of vectors

stipulates that "real numbers $\lambda$ with vector $\boldsymbol a$ "The product of" is a vector. This operation is the vector's Multiply operations, recorded as $\lambda \boldsymbol a$, its length and direction are specified as follows:

1. $|\lambda \boldsymbol a|=|\lambda||\boldsymbol a|$；
2. when $\lambda >0$ hour,$\lambda\boldsymbol a$ with $\boldsymbol a$ in the same direction, when $\lambda =0$ hour,$\lambda \boldsymbol a=\boldsymbol 0$,when $\lambda<0$ hour,$\lambda \boldsymbol a$ with $\boldsymbol a$ In the opposite direction.

According to the definition of number multiplication, we can verify that there are the following operational laws:

$$\lambda(\mu \boldsymbol a)=(\lambda \mu)\boldsymbol a\\ (\lambda+\mu)\boldsymbol a=\lambda \boldsymbol a+\mu \boldsymbol a\\ \lambda(\boldsymbol a+\boldsymbol b)=\lambda \boldsymbol a+\lambda \boldsymbol b$$

In particular, we have:

$$(-\lambda)\boldsymbol a=-(\lambda \boldsymbol a)=-\lambda(\boldsymbol a)\\ \lambda(\boldsymbol a-\boldsymbol b)=\lambda \boldsymbol a-\lambda \boldsymbol b$$

# Determine whether two vectors are collinear

two non-zero vector $\boldsymbol a$ with $\boldsymbol b$ collinear $\iff$ There is a unique real number $\lambda$, making $\boldsymbol b=\lambda \boldsymbol a$。

Proof: According to the definition of number multiplication, for non-zero vector $\boldsymbol a$, if there are real numbers $\lambda$, making $\boldsymbol b=\lambda \boldsymbol a$,So $\boldsymbol a \parallel \boldsymbol b$。

Conversely, if $\boldsymbol a\parallel \boldsymbol b$，$\boldsymbol a \not = \boldsymbol 0$,and $|\boldsymbol b|=\mu |\boldsymbol a|$, then when $\boldsymbol a$ with $\boldsymbol b$ When in the same direction,$\boldsymbol b=\mu \boldsymbol a$, when reverse $\boldsymbol b=-\mu \boldsymbol a$。

Finally, the addition, subtraction, and multiplication of vectors are collectively referred to as linear operations on vectors.

# Basic theorem and coordinate representation of plane vectors

# Fundamental Theorem of Plane Vectors

Theorem content: If two vectors $\boldsymbol{e_1},\boldsymbol{e_2}$ are not collinear, then there is a unique pair of real numbers $(x,y)$, such that $\boldsymbol{e_1},\boldsymbol{e_2}$ Coplanar arbitrary vectors $\boldsymbol p$ satisfy $\mathbf p=x\boldsymbol{e_1}+y\boldsymbol{e_2}$。

There are so many plane vectors, and we want to express all the plane vectors with as few quantities as possible. What should we do?

It is obviously impossible to express all vectors with only one vector. At most, it can only express vectors on a certain straight line.

We add another vector and use two Not collinear Vector representation (two collinear vectors can be regarded as the same vector here), so that we can decompose any plane vector into the directions of these two vectors.

Two non-collinear vectors in the same plane are called base。

If the bases are perpendicular to each other, then when we decompose the vector orthogonal decomposition。

# Coordinate representation of plane vectors

If we take the same unit vector as the horizontal and vertical axes, $i,j$ As a set of bases, according to the fundamental plane vector theorem, all vectors on the plane are equal to ordered pairs of real numbers $(x,y)$ One-to-one correspondence.

And ordered pairs of real numbers $(x,y)$ Corresponds one-to-one with the points on the plane rectangular coordinate system, then we make $\overrightarrow{OP}=\boldsymbol p$, then the end point $P(x,y)$ It's the only one that's certain. Since we are studying free vectors, we can freely translate the starting point. In this way, in the plane rectangular coordinate system, each vector can be uniquely represented by an ordered real number pair.

# Coordinate operations for plane vectors

# Plane vector linear operations

From the linear operation of plane vectors, we can deduce its coordinate operation. The main method is to convert all coordinates into base representations, and then merge them using the operation law, and then express the coordinate form of the operation result.

If two vectors $\boldsymbol a=(m,n)$，$\boldsymbol b=(p,q)$,but:

$$\boldsymbol a+\boldsymbol b=(m+p,n+q)\\ \boldsymbol a-\boldsymbol b=(m-p,n-q)\\ k\boldsymbol a=(km,kn)$$

# Find the coordinate representation of a vector

Two things are known $A(a,b),B(c,d)$, easy to prove $\overrightarrow{AB}=(c-a,d-b)$。

# Pan a little

Sometimes, we need to convert a point $P$ Translate a certain unit length in a certain direction, so that we combine the direction and distance to be translated into a vector, and use the triangle rule of vector addition to $\overrightarrow{OP}$ Adding this vector, the end point of the resulting vector is the translated point.

# Judgment of three points being collinear

If $A,B,C$ If three points are collinear, then $\overrightarrow{OB}=\lambda \overrightarrow{OA}+(1-\lambda)\overrightarrow{OC}$。

# Quantitative product of vectors

Two vectors are known $\boldsymbol a,\boldsymbol b$, their included angle is $\theta$,So:

$$\boldsymbol a \cdot \boldsymbol b=|\boldsymbol a||\boldsymbol b|\cos \theta$$

It’s these two vectors quantity product, also called dot product or inner product. which is called $|\boldsymbol a|\cos \theta$ for $\boldsymbol a$ in $\boldsymbol b$ projection in the direction. The geometric meaning of quantity product is: quantity product $\boldsymbol a \cdot \boldsymbol b$ equal $\boldsymbol a$ model and $\boldsymbol b$ in $\boldsymbol a$ The product of projections in the direction.

We found that the result of this operation is a real number, a scalar, and does not belong to the linear operation of vectors.

The quantity product operation has the following applications:

# Determine if two vectors are perpendicular

$\boldsymbol a \perp \boldsymbol b$ $\iff$ $\boldsymbol a\cdot \boldsymbol b=0$

# Determine whether two vectors are collinear

$\boldsymbol a = \lambda \boldsymbol b$ $\iff$ $|\boldsymbol a\cdot \boldsymbol b|=|\boldsymbol a||\boldsymbol b|$

# Coordinate operation of quantity product

If $\boldsymbol a=(m,n),\boldsymbol b=(p,q),$ rule $\boldsymbol a\cdot \boldsymbol b=mp+nq$

# vector module

$|\boldsymbol a|=\sqrt {m^2+n^2}$

# angle between two vectors

$\cos \theta=\cfrac{\boldsymbol a\cdot\boldsymbol b}{|\boldsymbol a||\boldsymbol b|}$

# Expand

# Vectors and matrices

(Elective 4-2 content of High School Mathematics Version A of People’s Education Press)

We found that the relevant laws of matrix operations are similar to vector operations, so we considered writing vectors in matrix form, thus turning the vector problem into a matrix problem.

Please refer to Linear Algebra for details.

# vector product

We define vector $\boldsymbol a,\boldsymbol b$ The vector product of is a vector, denoted as $\boldsymbol a\times \boldsymbol b$, its module and direction are defined as follows:

1. $|\boldsymbol a\times \boldsymbol b|=|\boldsymbol a||\boldsymbol b|\sin \langle \boldsymbol a,\boldsymbol b\rangle$；
2. $\boldsymbol a\times \boldsymbol b$ with $\boldsymbol a,\boldsymbol b$ are vertical, and $\boldsymbol a,\boldsymbol b,\boldsymbol a\times \boldsymbol b$ Complies with the right-hand rule.

Vector product is also called outer product.

Since vector product involves knowledge of space geometry and linear algebra, it does not appear in high school textbooks. However, notice the module of the vector product and think of the formula for calculating the area of ​​a triangle. $S=\frac{1}{2}ab\sin C$, we can find that the geometric meaning of vector product is:$|\boldsymbol a\times \boldsymbol b|$ Yes $\boldsymbol a,\boldsymbol b$ is the area of ​​the parallelogram with adjacent sides。

Knowing this, it is easy to calculate the area of ​​a polygon.

We have an incomplete coordinate representation: note $\boldsymbol a=(m,n),\boldsymbol b=(p,q)$, then the vertical coordinate of the vector product of the two vectors is $mq-np$, we can infer based on the right-hand rule and the vertical coordinate symbol $\boldsymbol b$ relative to $\boldsymbol a$ direction, if the vertical coordinate in the counterclockwise direction is a positive value, otherwise it is a negative value, abbreviated as Follow the negative and reverse the positive。

# vector rotation

Set $\boldsymbol a=(x,y)$, the inclination angle is $\theta$, the length is $l=\sqrt{x^2+y^2}$. but $x=l\cos \theta,y=l\sin\theta$. Make it rotate counterclockwise $\alpha$ degree angle, get the vector $\boldsymbol b=(l\cos(\theta+\alpha),l\sin(\theta+\alpha))$。

From the trigonometric identity transformation, we get,

$$\boldsymbol{b}=(l(\cos\theta\cos\alpha-\sin\theta\sin\alpha),l(\sin\theta\cos\alpha+\cos\theta\sin\alpha))$$

Simplify,

$$\boldsymbol b=(l\cos\theta\cos\alpha-l\sin\theta\sin\alpha,l\sin\theta\cos\alpha+l\cos\theta\sin\alpha)$$

put the above $x,y$ You can get it back on your behalf

$$\boldsymbol b=(x\cos\alpha-y\sin\alpha,y\cos\alpha+x\sin\alpha)$$

Even if you don't know the trigonometric identity transformation, this formula is easy to memorize.

# Polar coordinates and polar coordinate system

# Arbitrary angles and radians

We learned angle values ​​in junior high school, but angles are not a number, which brings certain difficulties to our in-depth research. There are also other problems that cannot be explained clearly, so we use the radian system to describe angles.

First, we use the idea of ​​rotation to define an angle. An angle can be viewed as a figure formed by a ray in a plane rotating from one position to another around its endpoint. The starting position is called the initial edge, and the ending position is called the final edge.

We stipulate that according to Counterclockwise The angle formed by directional rotation is called positive angle,according to clockwise The angle formed by directional rotation is called negative angle, if this ray does not perform any rotation, it is called zero angle. In this way we push the concept of angles to any angle。

Then we introduce radians, the central angle subtended by an arc whose length is equal to the radius is called $1$ Angle in radians, with symbol $\text{rad}$ Represents, pronounced as: radians.

Generally speaking, the number of radians of a positive angle is positive, the number of radians of a negative angle is negative, and the number of radians of a zero angle is $0$, if the radius is $r$ central angle of circle $\alpha$ The length of the arc subtended is $l$,but $|\alpha|=\frac{l}{r}$. Using this formula, we can also write the arc length and sector area formulas, which are skipped here.

Then, we find $360^\circ$ The angle in radians is $2\pi$, after having the corresponding relationship, we can convert the angle value into the radian system.

We consider an angle and rotate its end side one more time, or even multiple times, and the position of the beginning side does not change, then the position of the end side will always be the same. We call these angles Angles with the same terminal edge position。

with horns $\alpha$ The set of angles with the same terminal edge position is easily obtained as $\{\theta\mid \theta=\alpha+2k\pi,k\in \mathbb{Z}\}$。

It can be understood as: the edge of this angle is continuously rotated once, and the position of the final edge remains unchanged.