Question

Векторы и координаты

Answer 1

Вектор - это направленный отрезок. Обозначается вектор двумя способами: одной строчной латинской буквой(черточка сверху) или двумя заглавными латинскими буквами, где первая начало вектора, вторая конец вектора.
Каждая точка на плоскости (в пространстве) имеет две(три) координаты.
Плоскость:
пусть А(х₁,у₁) - начало вектора, В(х₂,у₂) - конец вектора, тогда вектор АВ имеет координаты (х₂-х₁,у₂-у₁).
Пространство:
пусть A(x₁,y₁,z₁) - начало вектора, В(x₂,y₂,z₂) - конец вектора, тогда вектор АВ имеет координаты (x₂-x₁,y₂-y₁,z₂-z₁)

Answer 2

Vectors and Coordinates

Vectors and coordinates are fundamental concepts in mathematics and physics. They are used to represent quantities such as position, velocity, and force in a geometric or physical space.

Vectors are mathematical objects that have both magnitude and direction. They can be represented by an arrow or a line segment. In a two-dimensional space, a vector can be represented by its x and y components, which are usually denoted as (x, y). The x component represents the horizontal displacement, and the y component represents the vertical displacement.

Coordinates are used to specify the position of a point in a space. In a two-dimensional space, coordinates are usually represented by a pair of numbers (x, y), where x represents the horizontal position and y represents the vertical position.

The relationship between vectors and coordinates can be understood as follows:

- A vector can be represented by the displacement between two points in a space. The coordinates of the starting point and the ending point of the vector determine its components. - The components of a vector can be used to determine its magnitude and direction. The magnitude of a vector is calculated using the Pythagorean theorem, and the direction is determined by the angle it makes with a reference axis.

For example, let's consider two points A(5, 1) and B(4, -3). The vector AB can be calculated by subtracting the coordinates of point A from the coordinates of point B:

AB = (x_B - x_A, y_B - y_A) = (4 - 5, -3 - 1) = (-1, -4)

Similarly, the vector BA can be calculated by subtracting the coordinates of point B from the coordinates of point A:

BA = (x_A - x_B, y_A - y_B) = (5 - 4, 1 - (-3)) = (1, 4)

Therefore, the vector AB is (-1, -4) and the vector BA is (1, 4).

In summary, vectors and coordinates are essential tools for representing and analyzing quantities in mathematics and physics. They provide a way to describe the magnitude and direction of quantities in a geometric or physical space.