In mathematics, we usually use points to indicate position and rays to indicate direction. On a plane, all rays emitted from any point can be used to represent all directions on the plane (Figure 5- 1).
In the fourth chapter "trigonometric function", we have learned the concept of "directed line segment", which only involves the directed line segment parallel to the coordinate axis. Generally, an order is specified between the two endpoints of the line segment AB. Assuming that A is the starting point and B is the end point, we say that the line segment AB has a direction, and the line segment with a direction is called a directed line segment (Figure 5-2). Usually, draw an arrow at the end of a directed line segment to indicate its direction. Record a directed line segment with a starting point and b ending point. Note: The starting point must be written before the end point. As we know, the length of line segment AB is also called the length of directed line segment, which is denoted as ||. A directed line segment contains three elements: starting point, direction and length. Knowing the starting point, direction and length of a directed line segment, we can uniquely determine its end point.
A vector is usually represented by a directed line segment, the length of which indicates the size of the vector, and the direction indicated by the arrow indicates the direction of the vector. As shown in Figure 5-3, the line segment with the length of 1cm indicates 5nmile, and the direction indicated by the arrow indicates the sailing direction of the ship, so the displacement of the ship in the chapter diagram can be represented by a directed line segment.
Vector can also be represented by letters A (1), b, c, etc. , or use letters to indicate the start and end points of vector directed line segments. For example, the vector (the displacement of the ship) in the chapter diagram can be expressed as A, etc.
The size of the vector, that is, the length (or modulus) of the vector, is expressed as ||. A vector with a length of 0 represents a zero vector, and a vector with a length equal to 1 unit length represents a unit vector.
Non-zero vectors with the same or opposite directions are called parallel vectors, as shown in Figure 5-4. A, B and C are a set of parallel vectors. Vectors A, B and C are parallel, which is called A ‖ B ‖ C. We stipulate that 0 is parallel to any vector.
Vectors with the same length and direction are called equal vectors. Vectors A and B are equal, so let's say A = B, and zero vector and zero vector are equal. Any two equal nonzero vectors can be represented by the same directed line segment, regardless of the starting point of the directed line segment.
As shown in Figure 5-4, if a straight line L is parallel to the straight line where A is located, it can be =a, =b, = C on L, that is, any group of parallel vectors can be moved to the same straight line, so parallel vectors are also called * * * line vectors.
For example, as shown in Figure 5-5, let O be the center of the regular hexagon ABCDEF, and the vectors are written as vectors, and respectively.
Solution:
Think about it, is the vector equal to? Does the vector equal?
practise
1. What is the length of a non-zero vector? How to express the length of non-zero vector? Are the two vectors equal in length? Are these two vectors equal?
2. Point out the length of each vector in the graph.
3.( 1) Two equal vectors are represented by directed line segments. If they start from the same place, do they end up at the same place?
(2) Two vectors with the same direction but different lengths are represented by directed line segments. If they have the same starting point, then do they have the same ending?
① Bold A for printing and writing.