1. ## Dot product

Question is in the attachment. Please open the attachment. Thank you.

2. 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

3. Hi Dan ,

Thank you very much for your reply. I tried to type the quetion into Word, unfortunately , it is not working properly.

Let me type the question again.

Question
If U*V = - IUI * IVI , and U and V are nonzero, then there exist a scalar k < 0 such that U = K*V .

Note: U and V are vectors.

=========================================
Here is what I can conclude so far:
If alpha is the angle between U and V , then cos alpha must = -1.
So alpha must = pi.
which implies that U and V point in the opposite directions.

My question is :
Is there existing a scalar K < 0 such that U = K*V when U and V point in the opposite directions?

I think the answer is Yes. K must be negative since they are pointing in the opposite directions.

Please let me know if I am wrong.

4. Originally Posted by Jenny20
Hi Dan ,

Thank you very much for your reply. I tried to type the quetion into Word, unfortunately , it is not working properly.

Let me type the question again.

Question
If U*V = - IUI * IVI , and U and V are nonzero, then there exist a scalar k < 0 such that U = K*V .

Note: U and V are vectors.

=========================================
Here is what I can conclude so far:
If alpha is the angle between U and V , then cos alpha must = -1.
So alpha must = pi.
which implies that U and V point in the opposite directions.

My question is :
Is there existing a scalar K < 0 such that U = K*V when U and V point in the opposite directions?

I think the answer is Yes. K must be negative since they are pointing in the opposite directions.

Please let me know if I am wrong.
Okay, I had missed that negative sign.

So there is only one possibility for your vectors: $\displaystyle cos(\theta_{UV}) = -1$ which means that $\displaystyle \theta_{UV} = 180^o$. Thus the two vectors are antiparallel (pointing in opposite directions.)

Thus U = k V where k is negative (to switch the direction.)

-Dan

5. Thus U = k V where k is negative (to switch the direction.)

-Dan[/QUOTE]

Thank you very much.