Simplification of Cross Product Expression

[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.

Thanks in advance

 

[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)

 

[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.

 

[Hurkyl]

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)

 

[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)

 

[robphy]

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

 

[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

 

[robphy]

Is the second term bc(u X v)?

 

[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 :)