1. Vector addition
If there are two vectors v = (v 1, v2, v3) and w = (w 1, w2, w3), their addition is defined as V+W = (V 1+W 1, V2+W2, V3+W3). That is to say, add the components at the corresponding positions to get a new vector.
2. Vector subtraction
If there are two vectors v = (v 1, v2, v3) and w = (w 1, w2, w3), their subtraction is defined as V-W = (V 1-W 1, V2-W2, V3-W3). That is to say, a new vector is obtained by subtracting the component of the corresponding position.
Multiply by numbers
Multiply a vector v = (v 1, v2, v3) with a scalar (real number) k, and the result of multiplication is kv = (kv 1, kv2, kv3). That is, each component of a vector is multiplied by a scalar.
4. Vector division
Vector division is not commonly used in general vector operations because the concept of division is not well defined in vector operations.
It should be noted that in vector operation, it is necessary to ensure that the vectors participating in the operation have the same dimension, that is, the number of their components is the same.
Definition of vector
Vector is a quantity with magnitude and direction, which is used to represent physical quantities such as displacement, force and speed in space. In mathematics, vectors are usually represented by ordered arrays or coordinates.
Generally speaking, a vector can be represented as an N-dimensional ordered array in an N-dimensional space, and each element is called a component of the vector. For example, in a three-dimensional space, a vector can be expressed as (x, y, z), where x, y, and z respectively represent the components of the vector on the x, y, and z axes.
A vector can be represented by an arrow, the direction of which indicates the direction of the vector, and the length of which indicates the size (or modulus or length) of the vector. Two vectors with the same size and direction are regarded as equal vectors.
Besides ordered arrays, vectors can be expressed in other ways, such as coordinate representation, decomposition representation, unit vector representation and so on. Different representations have their advantages and disadvantages in different contexts, but the basic concepts and attributes remain unchanged.
Vector is widely used in mathematics, physics, computer science and other fields, and is often used to describe and solve various problems, such as kinematics, mechanics, geometry and so on.
Use of addition, subtraction, multiplication and division of vectors
1. Vector addition
Vector addition can be used to calculate displacement, position change, velocity synthesis and so on. For example, in physics, if an object moves at a certain speed for a period of time, then changes direction and continues to move at another speed, the overall displacement and speed can be calculated by vector addition.
2. Vector subtraction
Vector subtraction can be used to calculate the difference vector, relative displacement, relative velocity and so on. For example, in navigation, if you need to calculate the relative displacement or direction between two places, you can use vector subtraction.
3. Digital Multiplication (Digital Multiplication)
Multiplication can be used to scale the size of a vector. By multiplying each component of a vector by a scalar, the size of the vector can be changed without changing its direction. This is very common in graphics rendering, calculations involving scaling and other applications.
4. Internal product and external product operation
The inner product and outer product of vectors can be applied to physics, geometry, engineering and other fields. The inner product can be used to calculate the projection, included angle and orthogonality of vectors, while the outer product can be used to calculate the cross product, area and vector operation of vectors.
It should be noted that the addition, subtraction, multiplication and division of vectors usually require that the vectors involved in the operation have the same dimension or meet specific operation rules. In addition, vector operations can also be used to solve linear equations, optimization problems and other mathematical and computational tasks.
Examples of vector addition, subtraction, multiplication and division
Suppose there are two vectors v = (2 2,3,4) and w = (1,-1, 2), and we use the above operation to calculate:
Vector addition: V+W = (2+ 1, 3+(- 1), 4+2) = (3,2,6).
Vector subtraction: V-W = (2- 1, 3-(- 1), 4-2) = (1, 4,2).
Number Multiplication: 2v = (2*2, 2*3, 2*4) = (4, 6, 8)
Multiplication: -0.5w = (-0.5 * 1, -0.5 *( 1), -0.5 * 2) = (-0.5, 0.5,-1).