What rules are satisfied by vector dot products and fork products associative laws, distributive la

VectorsFork multiplicationIt does not conform to the commutative law (b a direction facing down), conforms to the associative law, and the distributive law.

VectorsPoint multiplicationConforms to the commutative law, the associative law, and the distributive law.

Point multiplication is often used to: calculate the angle between two vectors; Calculate the projection of one vector on another; By the size of the angle, the similarity of the two vectors (similar direction, opposite, vertical, etc.) is judged.

The cross product of the vector results in a new vector perpendicular to the plane of ab, which conforms to the right-hand spiral rule.

Four fingers from A to B, with A B and the thumb in the same direction.

Apply. In production and life, dot product.

Wide range of applications. Use the dot product to determine whether a polygon is facing or facing away from the camera. The dot product of vectors and the cosine of their angles.

If the dot product is larger, the smaller the angle, the closer the physical axis to the axis of the light, the stronger the illumination.

In physics, the dot product can be used to calculate the resultant force and work. If b is the unit vector, then the dot product is the projection of a in the direction b, i.e., the decomposition of the force in this direction is given. Work is the dot product of force and displacement.

Computer graphics is often used to judge directionality, if the dot product of two vectors is greater than 0, their directions are similar; If it is less than 0, the direction is reversed. Vector inner product.

It is one of the mathematical foundations of neural network technology in the field of artificial intelligence, and this method is also used for animation rendering.

The fork multiplication does not satisfy the law of union.

The quantity product of the vector satisfies the associative law, i.e., (a·b)·c≠a· (b·c)； For example: (a·b) ≠a ·b.

The quantity product of the vector does not satisfy the elimination law, that is, from a·b=a·c(a≠0), b=c cannot be deduced.

a·b|with a|·|b|Not equivalent.

by a|=|b|A=b cannot be introduced, and a=-b cannot be introduced, but the reverse is true.

Mixed product. It has rotational symmetry.

a,b,c)=(b,c,a)=(c,a,b)=-a,c,b)=-c,b,a)=-b,a,c)。

In mathematics, vectors (also known as Euclid.

Vectors, geometric vectors, vectors) refer to quantities with magnitude and direction. It can be visualized as a line segment with an arrow. The arrow points to:

Represents the direction of the vector;Line Segment Length: Represents the size of the vector. The quantity corresponding to the vector is called the quantity (in physics, the quantity (or scalar) is only a size, and has no direction.

The two-dimensional vector fork multiplication formula a(x1,y1) trembling bucket, b(x2,y2), then a b (x1y2 x2y1), does not need to prove that Liang Yinming is the defined operation.

The three-dimensional cross-product is a determinant.

Operation, also a cross product.

The definition of the third dimension can be regarded as the substitution of 0 for the eggplant.

Algebraic rules1. Anticommutative law: a b=-b a

2. The distributive property of addition: a (b+c) a b+a c.

3. Compatible with scalar multiplication: (ra) b=a (rb) r(a b).

4. The law of association is not satisfied.

But satisfies the Jacobian identity: a (b c) b (c a) c (a b) 0.

5. The distributive property, linearity, and Jacobian identities show that R3 with vector addition and cross product constitutes a Lie algebra.

6. Two non-zero vectors a and b are parallel, if and only if a b = 0.

Because vectors have directions.

In mathematics, a vector (also known as a Euclidean vector, geometric vector, vector), refers to a quantity with magnitude and direction. It can be visualized as a line segment with an arrow. The arrow points to:

Represents the direction of the vector;Line Segment Length: Represents the size of the vector. The quantity corresponding to a vector is called a quantity (called a scalar in physics), and a quantity (or scalar) is only a magnitude and has no direction.

Notation of vectors: Letters (e.g., a, b, u, v) in bold (bold) are written with a small arrow on top of the letters" "If you give the beginning (a) and end (b) of the directional quantity, you can write the vector as ab (and add at the top). In the spatial straight-based Kai angle coordinate system, vectors can also be represented as pairs, for example, (2,3) in the xoy plane is a vector.

In physics and engineering, geometric vectors are more commonly referred to as vectors. Many physical quantities are vectors, such as the displacement of an object, the force exerted on a ball as it hits a wall, and so on.

The opposite is a scalar quantity, which is a quantity that has only a magnitude and no direction. Some of the fixed-beat potato meanings related to vectors are also closely related to physical concepts, such as the potential of vectors corresponds to potential energy in physics.

The concept of geometric vectors is abstracted in algebra to obtain a more general concept of vectors. Vectors are defined as elements of a vector space, and it is important to note that these abstract vectors are not necessarily represented by the number of pairs of watches, nor do the concepts of size and direction apply. Therefore, when reading on a daily basis, it is necessary to distinguish what is said in the text according to the context"Vectors"What kind of concept is it?

However, it is still possible to find the basis of a vector space to set up the coordinate system, and it is also possible to mediate the norm and inner product on the vector space by choosing the appropriate definition, which allows us to analogy vectors in the abstract sense to concrete geometric vectors.

Not satisfied, the direction after the fork is in line with the right-hand spiral rule.

1.The result of the vector cross multiplication is still a vector point multiplied by a number, the direction of this vector is judged by the right-hand spiral rule, the new vector after the cross multiplication is perpendicular to the original two, the four fingers turn from one vector to the other direction, and the direction of the thumb is the direction of the new vector.

2.According to the right-hand system, they represent vectors of equal magnitude and opposite directions, according to the definition of the vector envy product and the determination of its direction, this book and the encyclopedia certainly have.

3.The direction is different, the two vectors are multiplied together is a number, and the third vector multiplied is equivalent to extending the third vector by a factor of less, a*b*c is the direction of c, and a*(b*c) is the direction of a.

4.The left equation is equivalent to first calculating a·b, which is the product of the quantities of vector a and vector b, to obtain a constant, and then multiplying this constant by vector c to obtain a vector colinear with vector c.

5.The right equation is equivalent to calculating b·c first, which is the product of the quantities of vector b and vector c, and another constant is obtained, and this constant is multiplied by vector a to obtain a vector that is collinear with vector a.

6.Vector B is the same as vector C. But it is possible to shift the term and get a·b-a·c=0, and get a·(b-c)=0, i.e., the vector a is perpendicular to the vector (b-c), which is correct.

The vector cross multiplication does not conform to the commutative law (b a direction facing down), but conforms to the associative and distributive laws. The cross product of the vector results in a new vector perpendicular to the plane where ab is located, in accordance with the right-hand spiral rule, with four fingers from a to b, and a b and thumb in the same direction.

1. The scalar product of two vectors is a scalar quantity. The geometric meaning of the cross product is that the modulus of the result is the numerical product of the projection of a vector perpendicular to another vector, or the area of a parallelogram formed by two vectors as sides.

2. The result of the cross multiplication is still a vector, which is denoted as vector c. The product of a cross multiplication vector is a binary operation that takes two vectors on a real number r and returns a real-value scalar. It is the standard inner product of Euclidean space.

3. The direction of c is defined as the vector perpendicular to the plane of a and b and is determined by the direction of the vector space, i.e., according to the left and right hand rules of the given Cartesian coordinate system (x, y). If (x,y) satisfies the right-hand rule, then (x,y) also satisfies the right-hand rule; Or both satisfy the left-handed rule.

Algebraic rules: 1. Anticommutative law: a b=-b a

2. The distributive property of addition: a (b+c) a b+a c.

3. Compatible with scalar multiplication: (ra) b=a (rb) r(a b).

4. The associative law is not satisfied, but the Jacobian identity is full of strong feet: a (b c) + b (c a) + c (a b) = 0.

5. The distributive law, linearity, and the comparable identities of the ruler slag show that r3 with vector addition and cross product constitutes a Lie algebra.

6. Two non-zero vectors a and b are parallel, if and only if a b = 0. <>