Three-dimensional vector is a common concept in mathematics, which is used to represent ordered number pairs with three components. In three-dimensional space, we can use a directed line segment to represent a three-dimensional vector. The starting point of this directed line segment is called the origin and the end point is called the end point of the vector.

Three-dimensional vectors are usually represented by lowercase letters, such as (mathbf{v}). Its three components are expressed as (v_x), (v_y) and (v_z) respectively. These components can be real numbers or complex numbers.

There are many ways to express three-dimensional vectors, among which column vectors and row vectors are the most common. Column vector refers to arranging three components into a column in turn, for example, (begin {bmatrix} v _ XV _ yv _ zend {bmatrix}). Line vector refers to arranging three components into a line in sequence, such as (begin {bmatrix} v _ x&; v _ y & ampv_zend{bmatrix}).

Three-dimensional vectors have many important properties and applications. For example, it can be used to describe physical quantities such as position, speed and acceleration of an object. In addition, three-dimensional vectors can also be used to calculate geometric transformations, such as translation, rotation and scaling.