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Anonymous users2024-02-06The difference between the left-right distributive properties of vector forks and multiplications.
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Anonymous users2024-02-05ax(b+c)=axb + axc?
This can be brought in with the coordinates of the vectors a, b, and c, and the results are calculated on the left and right, respectively, and the equality is proved.
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Anonymous users2024-02-04The 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. <>
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Anonymous users2024-02-03The vector cross product does not satisfy the distributive law, and the direction after the fork conforms to the right-hand spiral law. 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.
According to the right-hand system, they represent vectors of equal magnitude and opposite directions, according to the definition of the vector product and the determination of its directional branch answer. The direction is different, multiplying the two vectors together is a number, and multiplying the third vector 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, so it is different.
The left formula is equivalent to calculating a·b first, which is the product of the quantities of vector a and vector b, and then multiplies this constant with vector c to obtain a vector that is collinear with vector c. 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.
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