Does dot product always commute?

[Pollywoggy]

I thought the dot product was commutative but there must be something about it that I don't understand. Perhaps the dot product is commutative only for vectors and not for tensors generally?

In Kusse and Westwig, p70, it says that the order of terms matters because, in general,

$$\hat{e}_j \hat{e}_k\cdot\hat{e}_l$$ does not equal $$\hat{e}_l\cdot\hat{e}_j\hat{e}_k$$

(the e's are all basis vector e's but I did not know how to show that)

[Kusse and Westwig, Mathematical Physics 2e (Wiley 2006)]

 

[Pollywoggy]

I just realized something, after posting my question. The reason the dot product is not generally commutative probably has to do with whether the vectors or tensors are orthogonal.
Is that it? I think what confused me was that the book uses subscripts j, k, and l. That should not have confused me but it probably did and I assumed orthogonality.

 

[guateca]

I do not know about tensors, but the dot product of vectors is commutative. However, the dot product of orthogonal vectors = 0

 

[kuruman]

I thought the dot product was commutative but there must be something about it that I don't understand. Perhaps the dot product is commutative only for vectors and not for tensors generally?

In Kusse and Westwig, p70, it says that the order of terms matters because, in general,

$$\hat{e}_j \hat{e}_k\cdot\hat{e}_l$$ does not equal $$\hat{e}_l\cdot\hat{e}_j\hat{e}_k$$

(the e's are all basis vector e's but I did not know how to show that)

[Kusse and Westwig, Mathematical Physics 2e (Wiley 2006)]

If your basis vectors are orthogonal, then the two expressions that you show are both zero, therefore equal. However, I am not sure what you mean by "dot product". If you mean "inner product" then it is strictly a scalar or zero rank tensor. The two expressions that you show are vectors (first rank tensors) pointing in different directions, one along "j" and the other along "k" so they are not generally equal.

 

[rwisz]

You are correct in your understanding that the dot product is commutative only for vectors and not for tensors in general. The dot product is a mathematical operation that takes two vectors and produces a scalar quantity, representing the projection of one vector onto the other. This operation is commutative for vectors because the order in which the vectors are multiplied does not affect the resulting scalar value.

However, when dealing with tensors, which are mathematical objects that can represent physical quantities with multiple directions and magnitudes, the dot product is not always commutative. This is because the order in which the tensors are multiplied can affect the resulting tensor, as shown in the example provided from Kusse and Westwig.

It is important to note that the commutativity of the dot product is dependent on the properties of the mathematical objects being multiplied. In the case of vectors, the dot product is commutative because they are both one-dimensional and have the same direction. But for tensors, the dot product may not be commutative if the tensors have different dimensions or directions.

In summary, the dot product is not always commutative, and this is a fundamental concept in mathematics and physics. It is important to understand the properties of the mathematical objects being multiplied in order to determine if the dot product will commute or not.