Expanding triple (cross) product

[leroyjenkens]

Homework Statement

Use (the bac-cab rule) to expand this triple product:L = mr x (ω x r)

If r is perpendicular to ω, show that you obtain the elementary formula, angular momentum = mvr.

(The bold letters are vectors.)

Homework Equations

A X (B X C) = (A$\cdot$C)B - (A$\cdot$B)C

The Attempt at a Solution

Well, simply doing the bac-cab rule, I get

(mr $\cdot$ω)r - (mr $\cdot$r)ω

Which isn't even close to the answer in the book, which is:

L=m[r²ω-(ω $\cdot$r)r]

No idea how they got that.

Even if I distribute the r, they don't have it distributed to the ω, and on the right term, they don't distribute the ω. I don't understand.
Thanks

 

[Abhilash H N]

The Attempt at a Solution

Well, simply doing the bac-cab rule, I get

(mr $\cdot$ω)r - (mr $\cdot$r)ω

This is wrong, going by the rule we get
m[(r.r)ω-(r.ω)r]
On simplification we'll get the answer...
Regards

 

[leroyjenkens]

This is wrong, going by the rule we get
m[(r.r)ω-(r.ω)r]
On simplification we'll get the answer...
Regards

OK yeah I lost track of which one was A, B, and C.

But this is what I get:
(mr$\cdot$r)ω-(mr$\cdot$ω)r

r$\cdot$r simplifies to r² because it's the vector dotted into itself, which produces the magnitude of that vector squared?

And according to the solution, why didn't they distribute the r through the parentheses in the second term?

Thanks.

 

[vela]

What specifically do you mean by "distribute the r through the parentheses"?

 

[leroyjenkens]

What specifically do you mean by "distribute the r through the parentheses"?

Oh ok, so that r isn't allowed to go into the parentheses until the r is dotted into the ω?

 

[vela]

I guess I still don't know what you're trying to do. $\vec{\omega}\cdot\vec{r}$ is a scalar, and the result of the product multiplies $\vec{r}$. By pulling $\vec{r}$ into the parentheses, what do you intend to accomplish?