# Simplification of Cross Product Expression

1. Nov 19, 2005

### vg19

Hey,

I have the following question,

Simplify
(au + bv) x (cu + dv) where a,b,c,d are scalars and u,v are vectors.

I know that we can take ab ab and cd outside to make the expression
ab(u +v) x cd(u + v) but I am unsure on where to go from here.

2. Nov 19, 2005

### ms. confused

I don't really know how you should go about simplifying this expression yet, but I do believe your first step is wrong.

See, the way you factored it, when you expand it again the expression would be : (abu + abv) x (cdu + cdv)

3. Nov 19, 2005

### robphy

You can't take ab outside (or cd outside), as you have done.
Instead, use the "FOIL" method, as you would for expanding out the ordinary product of two binomials (a+b)*(c+d). When doing this with the cross product, you have to keep order of the factors.

4. Nov 19, 2005

### Hurkyl

Staff Emeritus
What would you do if u and v were just numbers, and the cross product was just ordinary multiplication?

Can't you do most of the same thing in exactly the same way with cross products? (yes -- but you will have to pay attention to what won't work)

5. Nov 22, 2005

### vg19

Hmm. I thought of something else. Is this right?

(au+bv) X (cu+dv)
= (au X cu) + (au X dv) + (bv X cu) +(bv X dv)
= 0 + ad(u X v) + bc(u X v) + 0
= ad(u X v) + bc(u X v)

6. Nov 22, 2005

### robphy

almost.... can you justify the second equal sign?

7. Nov 22, 2005

### vg19

It would just be any vector crossed by itself gives 0. u X u = 0 and v X v = 0.

For the last line, can it be further simplified to

abcd(u X v)?

or should it remain ad(u X v) + bc(u X v)?

Thanks again

8. Nov 22, 2005

### robphy

Is the second term bc(u X v)?

9. Nov 22, 2005

### vg19

ohh bc(v X u)

So,

ad(u X v) + bc(v X u)

or

ad(u X v) -bc(u Xv) (because u X v = -(v X u) )

Hopefully I am now finally right :)