Given vector a=(2,0), b is a nonzero vector, if the angle between a+b, a-b and the positive direction of x axis is (∏/6) and (2∏/3), then b=

Given vector a=(2,0), b is a nonzero vector, if the angle between a+b, a-b and the positive direction of x axis is (∏/6) and (2∏/3), then b=

Let b=(x, y), then use the formula of included angle and simultaneous equations to find x, y.

Let b=(x, y), then use the angle formula and simultaneous equations to find x, y.

Why is the "product of vectors" a multiplication of moduli of two vectors and a multiplication of cos? Is this man-made? Or what?

The problem was introduced in terms of force and work.
Power = force * distance in force direction.
The problem is that forces and directions are vectors, and if there is an angle between the forces and the distance, then the product is not work.
Therefore, to project the force in the distance direction, the method is to multiply the cosine value of the included angle, so as to decompose the force in the distance, and then multiply the distance is the correct work.
Generalized to all vectors, just like that.

The problem was introduced in terms of force and work.
Power = force * distance in force direction.
The problem is that forces and directions are vectors, and if there is an angle between them, the product is not work.
Therefore, to project the force in the distance direction, the method is to multiply the cosine value of the included angle, so as to decompose the force in the distance, and then multiply the distance is the correct work.
Generalized to all vectors, just like that.

The problem was introduced in terms of strength and merit.
Power = force * distance in force direction.
The problem is that forces and directions are vectors. If there is an angle between the forces and the distance, then the product is not work.
Therefore, to project the force in the distance direction, the method is to multiply the cosine value of the included angle, so as to decompose the force in the distance, and then multiply the distance is the correct work.
Generalized to all vectors, just like that.

What is the geometric meaning of the vector product? Not a quantity product.

The operation and geometric meaning of the quantity product of plane vector is an important content of high school mathematics. It is widely used. It is a new operation after the addition and subtraction of vector and the product of real number and vector. It is a continuation of the previous knowledge. It is also the basis for learning the follow-up knowledge (such as the distance formula between two points, the positive and cosine theorems, the distance from point to straight line, etc.). It plays a connecting role. The quantity product of vector... It is easy to encounter obstacles when dealing with the vector angle whose starting point is not at the same point.The angle is the basis of the definition and geometric meaning of the number product of the learning vector.

Is the Vector Product Constant?

Yes

(Vector a + vector b) Why is the multiplied part of the expanded ^2 a product of quantities instead of a cross multiplication?

Cross multiplication is also called vector product. The result is a vector. It is generally used for the rotation of a physical rigid body. It is also used for the calculation of the third vector direction based on the judgment of the two vector directions. It is also used for the calculation of the vector product specified by the space vector in the geometry of the three-dimensional solution system in the university. Point multiplication is also called the quantity product. The result is...

Cross multiplication is also called vector product. The result is a vector. It is generally used for the rotation of a physical rigid body. It is also used for the calculation of the third vector direction based on the judgment of the two vector directions. It is also used for the calculation of the vector product specified by the space vector in the geometry of the solid solution system in the university. Point multiplication is also called the quantity product. The result is...

What is the difference between the number product of a vector and the multiplication of two vectors? Normally, what do two vectors OA.OB represent? For example, why isn't the multiplication of two vectors in the first sub- equal to |OA||OB|cosita?

The product of the number of vectors is the result of the multiplication of two vectors. To be exact, it is the result of the dot multiplication of two vectors. Just as the product is the result of the multiplication of two numbers. What do you mean by them?
There are two kinds of multiplication between vectors. In addition to the point multiplication mentioned above, there is another called cross multiplication. The result of cross multiplication is called [vector product], which is also called outer product, cross multiplication product; and [quantity product] can be called: inner product, point product. If you have not studied vector product, then you can completely equalize vector multiplication with quantity product.
As for the problem, as zddeng said:[ OA.OB] and [|OA OB cosθ], the two are basically equal, the latter is actually the former's definition, they are just the difference of form. When you know the definition of quantity product, you can convert them at will.
In fact,[ OA·OB] is only a notation of the product of the quantity of a vector. In order to obtain its result, it must be transformed according to the definition.[|OA OB cosθ] is a way of thinking, that is, to transform the multiplication of vectors into the multiplication of numbers and numbers. Another way of thinking is [coordinate method].
Otherwise you have to work out the length and angle of the vector according to the coordinates, and then use the length and angle to work out the product of quantity.