Is the Dot Product of Two Vector Pairs Always Commutative?

[digipony]

Homework Statement

The Attempt at a Solution

I am working a physics problem and want to make sure I'm not making a mistake in the math. Here is my math inquiry:

Say you have (a*b)(c*d) where * indicates the dot product, and a,b,c, and d are all vectors. Can you say that (a*b)(c*d) = (a*c)(b*d) since the dot product is commutative and gives you a scalar? Thanks!

 

[Dick]

Homework Statement

The Attempt at a Solution

I am working a physics problem and want to make sure I'm not making a mistake in the math. Here is my math inquiry:

Say you have (a*b)(c*d) where * indicates the dot product, and a,b,c, and d are all vectors. Can you say that (a*b)(c*d) = (a*c)(b*d) since the dot product is commutative and gives you a scalar? Thanks!

No. You can't. Neither of those properties says you can swap vectors between two different dot products.

 

[Karnage1993]

If you're ever unsure of these things, always try to find a counterexample. Take a = [1,0], b = [1,0], c = [1,1] and d = [1,2]. Then,

(a*b)(c*d) = (1)(1 + 2) = 3

but

(a*c)(b*d) = (1)(1) = 1

 

[Karnage1993]

Sorry, accidentally double posted.

 

[digipony]

No. You can't. Neither of those properties says you can swap vectors between two different dot products.

Darn- it made the problem so easy, but I guess that was a sign that I was probably doing something that I should not be doing. Thank you for explaining why I could not shuffle the vectors around. :)

If you're ever unsure of these things, always try to find a counterexample. Take a = [1,0], b = [1,0], c = [1,1] and d = [1,2]. Then,

(a*b)(c*d) = (1)(1 + 2) = 3

but

(a*c)(b*d) = (1)(1) = 1

I did try an example, however it worked out. But I guess it wasn't that good of an example since it was a fluke that it worked out. Thanks

 

[Ray Vickson]

Homework Statement

The Attempt at a Solution

I am working a physics problem and want to make sure I'm not making a mistake in the math. Here is my math inquiry:

Say you have (a*b)(c*d) where * indicates the dot product, and a,b,c, and d are all vectors. Can you say that (a*b)(c*d) = (a*c)(b*d) since the dot product is commutative and gives you a scalar? Thanks!

No, as others have already pointed out.

You can see this another way: a*b = |a||b| cos(a,b) and c*d = |c||d|cos(c,d), where |a|, |b| are the magnitudes of a and b and (a,b) is the angle between a and b, and similarly for c and d. In general, we do not have cos(a,c).cos(b,d) equal to cos(a,b).cos(c,d), so the two expressions are generally different.