Vector proof (cross product)

[Jbreezy]

Homework Statement

If x + v = u
Prove x x v = u x v = x x u

The Attempt at a Solution

I don't even know where to start with this. I thought that magnitude of the resultant vector would have to be equal. So I started messing with each to see if I could find a pattern.

x x v = | x|| v| sin θ

This is a crap approach I can't find anything. Please give me a hint not the answer. I just can't seem to draw the information in my text together to give myself enough to show this. Thank you.

 

[mfb]

Just use your definition of u in the equations, and simplify with known properties of the cross product.
It does not help to consider the magnitude of the vectors, as their "direction" has to fit, too.

 

[tiny-tim]

Hi Jbreezy!

Are you allowed to use the associative rule, a x (b + c) = a x b + a x c ?

Or do you have to use coordinates?

 

[Jbreezy]

I guess you are allowed to use whatever. I don't understand this problem at all.

 

[tiny-tim]

ok, then what is x x (u - v) ?

 

[Jbreezy]

0 vector? This is not anywhere in my eq. though. In terms of the original.

 

[Jbreezy]

This was a good problem to make me feel like a monkey with a stick.

so, If x + v = u
Prove x x v = u x v = x x u

Sub in u.

x x v = (x + v ) x v = x x(x + v)

So when you distribute.

X x V = X x V + V x V = X x X + X x v

So, V x V = 0 , X x X = 0
So
X x V = X x V = X x V
Thanks

 

[tiny-tim]

Hi Jbreezy!

(just got up :zzz:)

Yes, that's correct.

But it's a bit long-winded …

you could have done x x v = (x + v ) x v (because v x v = 0)

= u x v,​

or x x v - u x v = … (you finish it )