The problem of non zero vectors, a little incomprehensible

Because the non-zero vectors e1 and e2 are not collinear, the directions of e1 and e2 are different. The previous coefficients only change the modulus (i.e., length or distance) of the vector, so the two vectors are still unequal. For the two vectors to be equal, they need to be in the same direction as the modulus.

To (k- )e1=( k-1)e2 then k- =0 and k =1That is, for two non-collinear vectors to be equal, they must have a coefficient of 0The two zero vectors are equal.

k- and k-1 are equivalent to e1 and e2 being elongated or shortened, but e1 and e2 are not collinear, no matter how they are elongated or shortened, (k- )e1 cannot be equal to (k-1)e2, so in order to make them equal, they must be made to be zero vectors, that is, k- =0 k =1

If the non-zero vectors e1 and e2 are collinear, then there are non-zero real numbers such that e1 = e2

At this time. The non-zero vectors e1, e2 are not collinear and it is impossible to find non-zero real numbers such that e1 = e2

So to make (k-)e1=( k-1)e2 hold, only k- =0 k =1

The non-collinear ones should be equal, then only the zero vector can satisfy the equal sign, and there is no unified understanding of whether the zero vector is parallel to each other, and it is generally regarded as parallel.

The question is that e1e2 is not collinear and should be (k-)e1=( k-1)e2 If k- is not equal to 0 k is not equal to 1, then (k- )e1=( k-1)e2 can be e1e2 collinear and does not match the question so k- =0 k =1

Yes, because these two vectors are not collinear and the direction is not the same, but they are equal, so their coefficient has to be 0

You will only wait until it becomes a 0 vector.

It can't be zero.

0 is a scalar quantity. The result obtained by the vector product of a vector is a vector.

a×a=0。

The vector itself and its own fork product, the sine basis of the angle is 0, so a vector with a modulus length of 0 is 0, that is, a vector of 0, not 0.

The product of the quantity of the vector is the quantity.

And the result of the vector self and one's point product is the square of the modulus length, and you limit the modulus length to be non-0, so the result must not be 0.

History. Vectors, which were first applied to physics. Many physical quantities such as force, velocity, pulse-displacement, electric field strength, magnetic induction intensity.

and so on are vectors. About 350 BC, ancient Greece.

Famous scholar Aristotle.

Knowing that forces can be expressed as vectors, the combined action of two forces can be obtained using the famous parallelogram rule.

The term "vector" comes from directed line segments in mechanical, analytic geometry. The first to use directed line segments to represent vectors was the great British scientist Isaac Newton.

From the history of the development of mathematics.

For a long time in history, the vector structure of space was not recognized by mathematicians, and it was not until the end of the 19th century and the beginning of the 20th century that people associated the nature of space with vector operations, making vectors a set of mathematical systems with excellent generality of operations.

There's something wrong with those claims upstairs. Below I will explain them one by one.

1.The so-called vector, according to the definition of middle school, is a quantity with a size and a direction.

2.Vector parallelism is pointing in the same or opposite direction to the quantity.

3.Define the zero vector size to 0, and the direction can be arbitrary. In other words, it can be in the same direction as an arbitrary vector, i.e., parallel to an arbitrary vector.

4.There is no separate parallel vector, the parallel vector is mutual, a b, then a is a parallel vector of b, and b is also a. The landlord can ignore the word "non-zero" in the first sentence, which is because the person who compiled the high school textbook did not define the direction of the 0 vector before defining the vector parallel, which is actually the failure of the compiler.

To sum up, if we say a b, then we must first discuss whether a and b are zero, and as long as one of them is zero, then a and b must be parallel to each other, and then we must discuss the case that is not zero.

In addition, the sufficient and necessary condition of the proposition a b (b is not equal to 0) is that there is a unique Changshu k such that a = kb.

As long as the narrative is a parallel vector, then it must not be a zero vector, I think it is true.

Again, yes, he artificially prescribes that it is parallel (collinear) to any vector, but note that the direction of the zero vector is arbitrary.

It can only be said that zero vectors and any vector are collinear. , but it cannot be said that he is in the same direction as any vector.

For those who don't understand the above concepts, parallel is also called collinear.

A two-vector collinear means that the baselines are parallel or coincide.

That is, point a times b=0

0 vector special.

The unit vector of a non-zero vector has a certain direction and not necessarily a position.

In mathematics, vectors, also known as Euclidean vectors, geometric vectors, and vectors, refer to quantities with magnitude and skin orientation, which can be represented as line segments with arrows.

1. The arrow points to: represents the direction of the vector pants blindness;

2. Line segment length: represents the size of the vector.

Both the zero vector and the arbitrary vector are parallel, including itself.

The ultimate goal of studying vectors is to solve the model problem.

From the perspective of the spatial model of vectors, the so-called several zero vectors are actually different representations, and all the zero vectors in a space can be regarded as coincident.

Also, from an algebraic point of view, since the inner product of the zero vector and the zero vector is zero, you can also think of the zero vector and the zero vector as perpendicular; The outer product of the zero vector and the zero vector is also zero, so they can be seen as parallel. Zero vectors are both perpendicular and parallel to zero vectors.

So parallel vectors and vector parallels are not exactly equivalent.

Therefore, the textbook stipulates that parallel vectors are non-zero vectors, and the reason is to avoid the ambiguity you mentioned. 6. Don't blindly believe that reference books are written by people or copied from other people's ......Or what is the premise of this problem, 1, parallel vectors are non-zero vectors, what about zero vectors?

If vector A and vector b are parallel, and vector b and vector c are parallel, then are vectors a and c parallel?

If we talk about parallelism, how can we do 0 vectors? What if b is a 0 vector?

However, a reference book says that a and c are parallel vectors......

So what is the premise for this to be like this?

"This is a kind of regulation, to put it bluntly, mathematics cannot solve this problem, so such a provision is proposed."

For example: (1) If the product of two non-zero vector points is 0, it means that the two vectors are vertical, then the product of any non-zero vector point is 0, which means that the zero vector is perpendicular to any vector.

The zero vector modulus is 0 in any direction and can be parallel to any non-zero vector.

And. 2) Is it contradictory, in order to solve this problem, it is stipulated that the rotten zero vector is parallel to any vector.

Dude, don't think about it.

Answer: A Analysis: Selling Bright Gold: A vector is a quantity that has both magnitude and direction, so there is no vector.

There must be a direction, and Bu Huaixiang also stipulates that there is a zero direction parallel to any vector, so the zero vector is the only vector with an uncertain direction, so the proposition is wrong; For any vector a in the plane, as long as its modulo is 1, it is a unit vector.

From this it can be seen that the proposition is wrong; Collinear vectors.

That is, parallel vectors, including non-zero vectors and zero-type pulse vectors with the same direction or opposite directions, so the proposition is also wrong; Since an equality vector is a vector of equal length and the same direction, the proposition is correct; The correct answer is A

A vector manuscript with the same direction and length 1 is trapped.

Accordingly, for a non-zero vector v, its unit vector.

It should be. v/|v|

At the same time, it can be seen that the unit vector of the non-former respectful zero vector is unique.

For example: 3 5, 4 5) is in the same direction as (3, 4), and the former length = 1, so the former is the unit vector of the latter.

Question 1: Number of stupid sails:

Since it is stipulated that zero vectors are parallel to any vector, they cannot be said to be perpendicular.

Question 2: Yes, the direction of the zero vector is arbitrary.

But the direction of the zero vector is usually compared to another non-zero vector, and it will be slowed down so that the direction of the zero vector will be determined.

So there is a stipulation that the zero vector is parallel to any vector. )