A × (b × c) =

1. Mar 5, 2010

Shackleford

I worked on this every way I knew how last night, but I couldn't get it to work. I'm pretty sure it's something simple that I'm missing.

I've set u = (a x b), u = (c x d), etc.

2. Mar 5, 2010

Hurkyl

Staff Emeritus
Did you try the hint? Where did that lead you?

(where did u come from? )

3. Mar 5, 2010

Shackleford

Yeah, I tried the hint. I substituted u to put it in a triple scalar form. As far as I know, you can only manipulate three distinct vectors in the triple scalar. The (b x c) is a particular vector. That's why I substituted u in.

4. Mar 5, 2010

gabbagabbahey

It might be easiest for you if your first step is to use (ii) to calculate $(\textbf{a}\times\textbf{b})\times\textbf{c}$

5. Mar 5, 2010

Shackleford

Yeah, (a x b) x c = - c x (a x b). Dotting that identity with d explicitly gives you (i).

I tried to get (i) directly from (ii) by dotting a x (b x c) with d and manipulating the scalar product form. However, it seems that you're supposed to merely use the form of (ii) and its known identity to derive (i), not necessarily stick with the given (ii). Is that correct? Is it even possible to derive (i) the way I was going at it?

6. Mar 5, 2010

Hurkyl

Staff Emeritus
Well, what did you get? And did you try the other part of the hint, to rotate the factors of the triple?

7. Mar 5, 2010

Shackleford

I didn't get (i).

Yes. I rotated until I got back to where I started. I'm at work right now, and all my scratch work is at the house. I tried all variations of the triple scalar product.

8. Mar 5, 2010

Shackleford

(a x (b x c)) . d = (d x a) . (b x c)

(d x a) . (b x c) = c . ((d x a) x b) = b . (c x (d x a))

etc.

9. Mar 5, 2010

gabbagabbahey

You probably did derive (i) with your first attempt, you just didn't realize it.

The identity $(\textbf{d}\times\textbf{a})\cdot(\textbf{b}\times\textbf{c})=(\textbf{a}\cdot\textbf{c})(\textbf{b}\cdot\textbf{d})-(\textbf{a}\cdot\textbf{b})(\textbf{c}\cdot\textbf{d})$, is exactly the same thing as the identity $(\textbf{a}\times\textbf{b})\cdot(\textbf{c}\times\textbf{d})=(\textbf{a}\cdot\textbf{c})(\textbf{b}\cdot\textbf{d})-(\textbf{a}\cdot\textbf{d})(\textbf{b}\cdot\textbf{c})$...you simply have to rename your vectors $\textbf{a}\to\textbf{b}$, $\textbf{b}\to\textbf{c}$, $\textbf{c}\to\textbf{d}$ and $\textbf{d}\to\textbf{a}$.

Remember, vector identities are true for arbitrary vectors, so it doesn't matter how you name them.

10. Mar 5, 2010

Shackleford

And I was ready to throw in the towel, quit my major and minor and get a liberal arts degree. Well, that's just great. I knew it had to be something silly I was overlooking. The thought of renaming the vectors DID occur to me, but I didn't follow through with it. I spent an embarrassing amount of time on that problem manipulating the scalar product (ii) as written trying to get (i) as written. So, all I had to do was get the form of (i) FROM (ii) and change the vectors in the (ii) derivation to match the vectors in (i).

11. Mar 5, 2010

Shackleford

Well, I won't take it back yet. I really suck at this.

a × (b × c) = αb + βc

(αb + βc) · d = (αb · d) + (βc · d)

12. Mar 5, 2010

gabbagabbahey

Okay, so combining these two lines you have $[\textbf{a}\times(\textbf{b}\times\textbf{c})]\cdot\textbf{d}=\alpha(\textbf{b}\cdot\textbf{d})+\beta(\textbf{c}\cdot\textbf{d})$; now use the scalar triple product rule on the LHS to get something you can compare to (i).

13. Mar 5, 2010

Shackleford

I would imagine on the LHS I do what I did earlier??

(d x a) · (b x c) = α(b · d) + β(c · d).

If I do that, then I guess I could do

d to a
a to b
b to c
c to d

then α = (a · c), β = (b · c). IDK. I'm getting tired.

Last edited: Mar 5, 2010
14. Mar 5, 2010

gabbagabbahey

Right, and what does (i) tell you the LHS of this equation must be?

15. Mar 5, 2010

Shackleford

Did I get it right in the edit to my previous post?

16. Mar 6, 2010

gabbagabbahey

Close, but not quite. Try this one step at a time...what do you get when you apply (i) to $(\textbf{d}\times\textbf{a})\cdot(\textbf{b}\times\textbf{c})$?

17. Mar 6, 2010

Shackleford

Well, correspondingly,

(d x a) · (b x c) = (d · b)(a · c) - (d · c)(a · b) = α(b · d) + β(c · d)

(d · b)(a · c) - (d · c)(a · b) = α(b · d) + β(c · d)

α = (a · c)

β = -(a · b)