Originally Posted by
Jenny20 Question is in the attachment. Please open the attachment. Thank you.
The file is pretty messed up, but I think I got the question.
Code:
If two vectors $\displaystyle \vec{U}$ and $\displaystyle \vec{V}$ have the property that
$\displaystyle \vec{U} \cdot \vec{V} = \left | \vec{U} \right | \left | \vec{V} \right | $
then there exists a scalar $\displaystyle k \neq 0$ such that $\displaystyle \vec{U} = k \vec{V}$
(I changed the condition on k because the problem statement was not correct. k need not be negative.)
By definition:
$\displaystyle \vec{U} \cdot \vec{V} = \left | \vec{U} \right | \left | \vec{V} \right | cos \left ( \theta_{UV} \right )$
where $\displaystyle \theta_{UV}$ is the angle between the two vectors.
So the problem statement:
$\displaystyle \vec{U} \cdot \vec{V} = \left | \vec{U} \right | \left | \vec{V} \right | cos \left ( \theta_{UV} \right ) = \left | \vec{U} \right | \left | \vec{V} \right |$
means that
$\displaystyle cos \left ( \theta_{UV} \right ) = 0$
which means that $\displaystyle \theta_{UV}$ = 0 or 180 degrees. Thus the vectors are parallel or antiparallel.
Vectors which are parallel (or antiparallel) are the same vector, up to a rescaling, ie. there exists a scalar k (which just changes the length of the vector) such that $\displaystyle \vec{U} = k \vec{V}$. (k is positive if the vectors are parallel and negative if they are antiparallel.)
-Dan