Why is scalar product commutative while vector product is not?

If you apply the right-hand rule, the vector points in the opposite direction as before. It points below the plane defined by and . This is precisely why is not commutative: and have opposite directions.

Are scalar and vector products commutative?

The scalar product of two vectors is commutative.

Why is the scalar product commutative?

The dot product of a vector with itself is the square of its magnitude. The dot product of two vectors is commutative; that is, the order of the vectors in the product does not matter. Multiplying a vector by a constant multiplies its dot product with any other vector by the same constant.

Why the cross product of two vectors A and B is not commutative?

The cross product does not follow the commutative property because the direction of the unit vector becomes opposite when the vector product occurs in a reverse manner. Hence, both the cross products of both the vectors in both the possible ways. i.e. AxB and BxA are additive inverse of each other.

What is scalar product and vector product?

Scalar products and vector products are two ways of multiplying two different vectors which see the most application in physics and astronomy. The scalar product of two vectors is defined as the product of the magnitudes of the two vectors and the cosine of the angles between them.

What is difference between scalar product and vector product?

If the product of two vectors is a scalar quantity, the product is called a scalar product or dot product. If the product of two vectors is a vector quantity then the product is called vector product or cross product. If two vectors are perpendicular to each other then their scalar product is zero.

Does scalar product and vector product obey associative law?

Notice that the dot product of two vectors is a scalar, not a vector. So the associative law that holds for multiplication of numbers and for addition of vectors (see Theorem 1.5 (b),(e)), does not hold for the dot product of vectors.

Why the cross product is not commutative?

We must note that only the direction of the vectors a×b and b×a are different, while the magnitudes of the two are equal. The opposite directions of the two vectors make the cross product non-communicative.

Why is it important to distinguish between scalar and vector?

A scalar quantity has only magnitude, but no direction. Vector quantity has both magnitude and direction. Any mathematical operation carried out among two or more scalar quantities will provide a scalar only. …

Is scalar and vector multiplication commutative?

Scalar multiplication is the product of a scalar and a vector- you can’t interchange them. Of course, if just want to say that it doesn’t matter how you write the product where is a scalar and v is a vector, then that’s trivially true but that is not what “commutative” means!

What is the scalar product of two vectors?

The scalar Product: Product Of Two Vectors. Many physical quantities we deal with are represented as vector quantities, such as velocity, force etc. These quantities interact with each other to produce a resultant effect. In order to find the resultant of these forces, operations such as addition, subtraction and multiplication are required…

Is the dot product of two vectors always commutative?

Fact 1 implies that the dot product is commutative: reversing the order of the vectors does not alter or their lengths. Trying to apply the same reasoning to Fact 2 only tells us that and have the same length. But Fact 2 also forces the cross product of a vector with itself to always be , because then , so . . .

Is cross product commutative or not?

Indeed not only cross product is not commutative, further it is anticommutative. See, also, the cross product tag wiki. Thanks for contributing an answer to Mathematics Stack Exchange!