How to learn mathematical vectors well

First, the concept of vector

The quantities we encounter in our daily life can be divided into two categories: one can be completely expressed by a numerical value, such as area, temperature, time or mass, which is called quantity (or scalar); Another kind of quantity, except a number, can only be fully expressed by indicating its direction, such as speed, acceleration, force and so on. , belong to this category.

Is a vector (or vector).

Vectors can be represented intuitively by directed line segments. The direction of the line segment indicates the direction of the vector, and its length is called the module of the vector. A vector is often written as (a→), (b→) or a,

B and so on Sometimes a vector is represented by (A→B), where a is the starting point and b is the end point. The direction from a to b indicates the direction of (a→). The modulus of vector (A→B) is |(A→B)|. A vector with a modulus equal to zero is called a zero vector, and it is recorded as 0 or (0→). The direction of the zero vector can be considered arbitrary. A vector whose modulus is equal to 1 is called a unit vector. For non-zero vector (a→), we use (a (0 →)) to represent the unit vector in the same direction as A, which is referred to as the unit vector of A for short. In Cartesian coordinate system, the vector (O→M)

It is called the radial direction of point M, and it is denoted as r or (r→).

. So every point m in space corresponds to a radial direction.

; On the other hand, each radial R corresponds to a certain point M. When two vectors have the same direction and the same modulus, they are called equal vectors and recorded as (a→).

=(b→)

. Therefore, a vector is equal to the original vector after translation. A vector with the same modulus and opposite direction is called

The negative vector of, expressed as (a→)=-(c→).

Second, the vector and operation

1, vector addition

Two vectors (O→A)

The sum of (O→B) is the diagonal vector (O→C) of a parallelogram with these two vectors as adjacent sides.

, recorded as (O→A)+(O→B)=(O→C)

This method is called parallelogram rule of vector addition. Because the opposite sides of the parallelogram are parallel and equal, we can also do the sum of two vectors: do.

(O→A)=(a→). Starting from the end of (a→) (b→)=(A→C)

, connecting OC

, you have to (O→C)

. This method is called the triangle rule of vector addition. The addition of vectors satisfies the commutative law and associative law. If there is a vector (a→)

，(b→)

That is, (a→)+(b→)=(b→)+(a→)

[(a→)+(b→)]+(c→)=(a→)+[(b→)+(c→)].

In particular, if (a→)

And (b→)

* * * straight lines (parallel or on the same straight line), and the sum of them is this vector: when (a→).

And (b→)

When their directions are the same, the direction of the sum vector is the same as that of the original two vectors, and its module is equal to the sum of the modules of the two vectors; When (a→)

And (b→)

When their directions are opposite, the direction of the sum vector is the same as that of the longer vector, and the modulus is equal to the modulus of the larger vector minus the modulus of the smaller vector.

2. Vector subtraction

Subtraction is the inverse of addition, if (b→)+(c→)=(a→)

The definition of (c→)

Is the vector (a→)

And (b→)

The difference is recorded as (c→)=(a→)-(b→).

Because (a→)+[-(b→)]=(a→)-(b→)

So we can get the corresponding laws of subtraction from the law of addition: (a→) and -(b→).

If the adjacent sides are parallelograms, the diagonal vector is (c→).

. If (a→)

And (-b→)

The starting point is the same, from (b→)