What does vector addition mean and what is vector addition?

Well, let's talk about the addition and subtraction of vectors:

In the case you said, vectors are generally represented by 2 uppercase letters, or by a lowercase letter.

If a represents the vector ab and b represents the vector ac, then: a+b=ab+ac, which means ab and ac.

A diagonal line of a parallelogram on an adjacent side, point A is the common starting point.

b-a=ac-ab=bc represents another diagonal of a parallelogram adjacent to ab and ac.

Note: Vectors must be distinguished from scalars, a=1cm, b=2cm, do a+b.

Take a little o, oa=a, ab=b, ob=a+b. Is OAB as many degrees as it takes? -- There is a problem with this statement.

What do you mean:|oa|=1，|ab|=2，|ob|=3, if you only consider the size of the vector, regardless of the direction, it is not right.

Nor do you come to the conclusion, the correct conclusion is: the vector ab=ob-oa, in the form of |oa|、|ob|、|ab|for 3 sides.

Triangle OAB, |ab|=|ob-oa|, and you don't say, "Then why is the sum of the two sides equal to the third side?" "This result.

If ab and ac are vectors, then they are wrong. The vector addition should consider the direction of AB and AC, and how to add it will not be BC.

Yes, explaining in physical language is actually the displacement from the initial position to the last position.

Vector addition is the operation of finding the sum of two or more vectors. The addition of vectors is connected end to end, that is, the starting point of the second vector is connected to the end point of the first vector, and the result is that the starting point of the first vector and the last end point are taken. That is, vector ab + vector bc = vector ac.

The direction of the line segment is the direction from one point to the other, then the two ends of the line segment have an order, we call the previous point the starting point noisy, the other point is called the end point, and draw an arrow at the end point to indicate its direction.

Geometric significance of vector additionVector addition in geometry is defined by geometric graphing. There are generally two methods, namely the triangle rule of vector addition and the parallelogram rule.

The textbook uses the triangular ascending source liquid shape rule to define it, which is also applicable to the collinear of two vectors, and the triangular rule of vector addition is consistent with the parallelogram rule when the vector is not collinear.

The addition and subtraction of vectors are as follows:

To put it simply: the addition and subtraction of vectors is the addition and subtraction of the corresponding components of vectors, and the coordinates of the sum and difference of the two vectors are equal to the sum and difference of the corresponding coordinates of the two vectors, respectively, if the vector is expressed in the form (x, y). The details are as follows:

Addition of vectors: a+b=(x1+x2,y1+y2).

Subtraction of vectors: a-b = (x1-x2, y1-y2).

The addition of vectors satisfies the parallelogram rule.

and the law of triangles disturbs the stool; The addition, subtraction, and multiplication of vectors (vectors have no division) satisfy the rough rule of addition, subtraction, and multiplication of real numbers.

Vector addition and subtraction rules:

The triangle rule.

The triangle rule solves the vector addition method: connect the vectors one after the other, and the result is that the starting point of the first vector is slowly stopped and compared to the end of the next vector.

The parallelogram rule.

The parallelogram rule solves the vector addition method: translate the two vectors to a common starting point, and make a parallelogram with the two sides of the vector, and the result is the diagonal of the common starting point.

Vector addition is easier to understand according to the method of physics, you can see two vectors as two directions to do the force, then their resultant force is the addition of two vectors (The parallelogram rule

Vector addition can be done using the parallelogram rule and the triangle rule, and if the starting point coincides, use the parallelogram rule, or the triangle rule, where the end is connected, and the starting point points to the end.

Calculation tips for vectors:

Medium, vector pulse-sensitive modulus.

It is usually represented by adding two vertical bars on each side of the vector, such as ||v||, representing the modulo of the vector v. For many vectors, we don't need to focus on its size, we only need to care about its direction, in this case we use unit vectors.

It will be very convenient. A unit vector is a vector of size 1, and a unit vector is also known as a normalized vector. For any non-zero vector v, a unit vector n in the same direction as v can be computed, a process called "normalization" of the vector.

The geometric meaning of vector addition is to connect the vectors one after the other, with the result that the starting point of the first vector points to the end point of the last vector.

Vectors are the bridge to transform geometric problems into algebraic problems, and the addition and subtraction of vectors is the use of algebraic methods for geometric operations

The triangle rule solves the vector addition method: the vectors are connected one after the other, and the result is that the starting point of the first vector points to the end of the last vector.

Parallelogram rule solution.

The method of determining the addition of vectors: translate the two vectors to the common starting point, and make a parallelogram with the two sides of the vector Qingzha, and the result is the diagonal of the common starting point.

The parallelogram rule solves the vector subtraction method: translate the two vectors to the common uproar early point, and make a parallelogram with the two sides of the vector, and the result is from the end point of the subtractive vector to the end point of the subtracted vector (the parallelogram rule only applies to the addition and subtraction of two non-zero non-collinear vectors).

You can follow these steps when learning vectors:

1. Learn the definition of vectors: vectors are quantities with size and direction, which are mathematically represented by arrows, the length of the arrows indicates the size of the vector, and the direction of the arrows indicates the direction of the vector.

2. Learn the basic operations of vectors: addition, subtraction, number multiplication, point multiplication, etc. Addition and subtraction are performed by adding and subtracting the length and direction of vectors.

Number multiplication is done by multiplying the length of the vector by a real number with the same direction. Point multiplication is done by adding the product of the lengths of two vectors to get a scalar.

3. Draw a graphical representation of the vector: you can use the arrow to represent the vector, the length of the arrow indicates the size of the vector, and the direction of the arrow indicates the direction of the vector, so that the vector can be represented more vividly.

4. Learn the coordinate representation of vectors: you can use coordinates to represent vectors, a two-dimensional vector can be represented by two numbers, and a three-dimensional vector can be represented by three numbers.

1. Addition of vectors: satisfies the parallelogram rule.

and the law of triangles, ie.

2. Subtraction of vectors: If a and b are opposite vectors, then a=-b, b=-a, a+b=0The inverse of 0 is 0oa-ob=ba

i.e. "common starting point, pointing to the subtracted", e.g. a=(x1,y1),b=(x2,y2), then a-b=(x1-x2,y1-y2).

3. Multiplication of vectors: the cross product of real numbers and vector a.

The product is a vector quantity, denoted as a, and |λa|=|a|。When >0, the direction of a is the same as that of a; When <0, the direction of a is opposite to that of a; Jujube and when =0, a=0, the direction of the stool is slow to stare. When a=0, there is a=0 for any real number.

4. Vector division: a k=|a|Vector of units of k*a.

That is, the result is the vector after the length of the original vector Nabu is reduced k times, and the direction remains unchanged.

Extended Information: 1. Arithmetic Law of Vector Addition:

1. Commutative property: a+b=b+a;

2. Associative law.

a+b)+c=a+(b+c)。

3. Addition and subtraction transformation law: a+(-b)=a-b

4. Addition and subtraction of vectors.

Multiplication (vectors without division) satisfies the rule of addition, subtraction, and multiplication of real numbers.

2. The law of number multiplication of vectors:

1. The quantity product of the vector does not satisfy the associative law, i.e., (a·b)·c≠a· (b·c)； For example: (a·b) ≠a ·b.

2. The quantity product of the vector does not satisfy the elimination law, that is, from a·b=a·c(a≠0), b=c cannot be deduced.

Let the vectors a=(x1,y1),b=(x2,y2), then a+b=(x1+x2,y1+y2).

Arithmetic of vector addition:

Commutative law: a+b=b+a

Associativity: (a+b)+c=a+(b+c).

In the Cartesian coordinate system, the original mu is defined as the starting point of the vector, and the coordinates of the sum and difference of the two vectors are equal to the sum and difference of the corresponding coordinates of the two vectors, if the vector is expressed in the form of (x,y), a(x1,y1) b(x2,y2), then a+b=(x1+x2,y1+y2).

Various graphical rules solve vector addition and subtraction.

1. The method of solving vector subtraction by the triangle rule: connect the vectors of each difference kernel in turn, and the result is that the starting point of the first vector points to the end point of the last vector.

2. The method of solving the vector addition method of the parallelogram rule: translate the two vectors to the common virtual excavation starting point, and use the two sides of the vector as a parallelogram, and the result is the diagonal of the common starting point.

Vector addition is a basic linear algebraic operation

It is closely related to our daily lives. In mathematics and physics, vector addition is widely used in vector motion, mechanics, electromagnetism, and other fields. In computer science, vector addition is also widely used in fields such as graphics, computer vision, machine learning, and artificial intelligence.

In two-dimensional space, the definition of vector addition is relatively simple and intuitive, and any vector can be expressed as the sum of two scalar quantities (real numbers) multiplied by two basis vectors.

Similarly, the addition of two vectors is the addition of the scalar quantities of their corresponding positions to get a new vector. For example, if there are vectors a=(x1,y1) and vectors b=(x2,y2), then their vector additions are: a+b=(x1+x2,y1+y2).

The addition operator here does not refer to ordinary addition, but to vector addition. As you can see, the result of vector addition is a new vector, whose direction and magnitude are determined by the original two vectors. On a two-dimensional plane, the effect of vector addition is to connect the tail of two vectors to get a new vector.

Similarly, in 3D space, vector addition can be defined in a similar way. Any vector can be expressed as the sum of three scalar multiplications by three basis vectors, and the cryptolic addition of two vectors is also the addition of the scalar at their corresponding position to get a new vector.

For example, if there are vectors a=(x1,y1,z1) and vectors b=(x2,y2,z2), then their vector additions are: a+b=(x1+x2, y1+y2,z1+z2) It can also be seen that in three-dimensional space, the result of vector addition is also a new vector, and its direction and size are also determined by the original two vectors.

It is important to note that vector addition has a commutative and associative law. That is, the addition of any two vectors can be swapped order, and the Eucha addition of multiple vectors can change the order of parentheses combined, and the result will be the same. For example:

a+b=b+a；(a+b)+c=a+(b+c)

The geometric significance of vector addition is important because it provides us with a simple, intuitive way to describe motion and mechanical phenomena in the physical world. In practical applications, vector addition is also often used to solve various problems, such as solving rigid body equilibrium, motion states, circuit analysis, etc.

At the same time, vector addition is also one of the indispensable basic operations in the fields of computer graphics, computer vision, machine learning, and artificial intelligence.

As far as the vertical friend god is the triangle rule of the residual loss vector addition and the parallelogram rule.

1) The law of triangles.

Vector oa + vector ab = vector ob

2) The parallelogram rule.