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- Nov 30th 2006, 04:48 PMJenny20Dot product
Question is in the attachment. Please open the attachment. Thank you.

- Nov 30th 2006, 05:36 PMtopsquark
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 $\displaystyle \vec{U} = k \vec{V}$

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 - Nov 30th 2006, 06:02 PMJenny20
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.

The answer is TRUE.

=========================================

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. - Nov 30th 2006, 06:16 PMtopsquark
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 - Nov 30th 2006, 06:21 PMJenny20
Thus U = k V where k is negative (to switch the direction.)

-Dan[/QUOTE]

Thank you very much.