Vector operation


vector (English: vector , physics, engineering, etc., also known as vector ) It is a basic concept in multiple natural sciences such as mathematics, physics and engineering science, refers to a geometric object that has a size and direction and satisfies parallel quadramph. In general, it is considered to be vectors, while satisfying the two properties of the two properties, it can be considered to be vectors (in particular, the current is both size, and there is an amount of positive and negative direction, but because the calculation does not satisfy the parallelogram method, it is recognized as It doesn't belong to the vector). Vector is often marked with a symbolic arrow to distinguish between other quantities. With the contained concept of the vector, the scalar or number , that is, only size, there is no direction in most cases (current is a special case), which does not satisfy the parallelogram method.

The size of the vector is relatively, and when there is a need, the unit vector is specified, and the length is used as 1. There is a unit vector in each direction.

The vector can be operated as a number. Common vector operations are: addition, subtraction, number multiplier, and multiplication between the vectors (quantity and vector).

addition and subtraction

The addition of the vector is satisfied with the parallelogram method and the triangular rule. Specifically, the two vectors A and B are added, and the other vector is obtained. This vector may be represented as a diagonal, a parallel tetragonal shape composed of A and B, or represents the end point of the starting point from A. After the starting point of the endpoint and B, the end point of the start point from A. Vector:

The two vectors A and B are subtracted, and it can be seen as a vector A plus a vector equal to the B size and the direction opposite. Alternatively, the decreased vector of A and B can be represented by the start of A and B, and the end point of the A and B points from the end point of the endpoint of A. The vector:

When these two vector values, direction Different, the basic vector

, vector and calculated as

and there is a different relationship:

In addition, the plus of the vector also meets the exchange law and the law.

vector and product

vector space is divided into finite dimensional direction and unlimited vector space. In a finite dimensional space, a set of (limited) vectors

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Vector operation

The scalar

is determined with the vector V. Such a set of vectors is called the basis of the vector space. The vector space and a set of batches are given, each vector can be expressed in an array. The two vectors V and W are identical when and only their arrays are indicated.

two vector V and W and W:

Their number is:

and the number of points k and vector v Then:

Scalar multiplication

A scalar k and one vector V can Make multiplication, the result is the same or in contrast to the V direction, the size of the size of the size of V, can be recorded, can be recorded as

. This calculation is called Scheduling Multiplication or number by . -1 multiplied by any vector, and 0 multiplied by any vector will get zero vector 0.

Number of products

The number of points is also called a point, which is the product of the vector and the product, the result is a scalar (non-vector). In the geometry, the number of columns can be defined as follows:

set A, B is two orientation, their angle is

, their number of quantities:

, an a vector in the B vector direction (the same direction is a positive and reverse direction), the product of the length of the B vector length. The number of products is widely used in physics, such as doing work, is the vector multiplying power of force, that is,


to quantify the quantity, the quantity, which is also the product of the vector and the vector, but it should be noted that its result is a vector. Its geometric meaning is that the resulting vector is vertical with the multiplying vector, and the direction is defined by the right hand, and the size is the area of ​​the parallelogram of the multi-ride amount. Therefore, it is not satisfied with the quantity. For example,


is provided with vector


, the quantity matrix expression can be represented by the following symbols:

Mixed Volumes

The mixing volume of the three vectors A, B, and C is defined as the volume consisting of three natives begins with the same time:

conditions are: A, B, and C vector composition of the right hand, is the positive number.

Considering and vector

must clearly distinguish between the market. See the table below.

< Td width = "141"> Geometric meaning
Name Time / Introduction / Quantity Level / Dots arithmetic formula (A, B and C bold characters, representation vector) a · B = | a || b | · cos θ a × B = c , of which | c | = | a || B | · SIN θ, c Observe the right hand
vector a in vector B direction projection with vector B Mode of the model c is equal to a and b as The parallel quadramographic area of ​​the neighbor scalar (commonly used for physical) / number of computation results scalar (Commonly used in mathematics) vector (commonly used in physical) / vector (commonly used in mathematics)
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