- #1
mitch987
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Help!? Linear algebra proof
Suppose that u,v,w are geometric vectors such that u[tex]\neq[/tex]0,
u[tex]\cdot[/tex]v=u[tex]\cdot[/tex]w and uxv=uxw
Prove that v=w
So far, I'm not sure if this is correct
u[tex]\cdot[/tex]v=u[tex]\cdot[/tex]w
|u||v|cos[tex]\theta[/tex]=|u||w|cos[tex]\theta[/tex]
|v|=|u|uxv=uxw
|u||v|sin[tex]\theta[/tex][tex]\hat{(u\times v)}[/tex]=|u||w|sin[tex]\theta[/tex][tex]\hat{(u\times w)}[/tex]
|w|sin[tex]\theta[/tex][tex]\hat{(u\times v)}[/tex]=|w|sin[tex]\theta[/tex][tex]\hat{(u\times w)}[/tex]
[tex]\hat{(u\times v)}[/tex]=[tex]\hat{(u\times w)}[/tex]
therefore, v=w
Homework Statement
Suppose that u,v,w are geometric vectors such that u[tex]\neq[/tex]0,
u[tex]\cdot[/tex]v=u[tex]\cdot[/tex]w and uxv=uxw
Prove that v=w
Homework Equations
The Attempt at a Solution
So far, I'm not sure if this is correct
u[tex]\cdot[/tex]v=u[tex]\cdot[/tex]w
|u||v|cos[tex]\theta[/tex]=|u||w|cos[tex]\theta[/tex]
|v|=|u|uxv=uxw
|u||v|sin[tex]\theta[/tex][tex]\hat{(u\times v)}[/tex]=|u||w|sin[tex]\theta[/tex][tex]\hat{(u\times w)}[/tex]
|w|sin[tex]\theta[/tex][tex]\hat{(u\times v)}[/tex]=|w|sin[tex]\theta[/tex][tex]\hat{(u\times w)}[/tex]
[tex]\hat{(u\times v)}[/tex]=[tex]\hat{(u\times w)}[/tex]
therefore, v=w