Verifying Vector Formula: A.B = B.A | Simple Homework Solution

[BillMath]

Homework Statement

Hi all my new friends..
A and B are vectors, A.B = B.A

Homework Equations

The Attempt at a Solution

How we can veryfy that this formula is correct?

 

[Vykan12]

I'm assuming you mean a scalar product.

A.B = ABcos(x)
B.A = BAcos(x)

AB = BA since arithmetic multiplication is commutative.

 

[tomcue]

This formula is known as the commutative property of vector multiplication. It states that the order of multiplication does not affect the result when multiplying two vectors together. To verify this formula, we can use the dot product definition of vector multiplication, which states that A.B = |A||B|cosθ, where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them.

Using this definition, we can see that A.B = |A||B|cosθ = |B||A|cosθ = B.A. This shows that the order of multiplication does not affect the result, and therefore the formula A.B = B.A is correct.

We can also visualize this concept using vector diagrams. When we multiply two vectors together, the result is a scalar quantity (a number), which represents the projection of one vector onto the other. This projection is independent of the order in which the vectors are multiplied, as shown in the diagram below.


Therefore, the commutative property holds true for vector multiplication, and the formula A.B = B.A is a valid and verified formula.