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- November 30th 2006, 05:48 PMJenny20Dot product
Question is in the attachment. Please open the attachment. Thank you.

- November 30th 2006, 06:36 PMtopsquark
The file is pretty messed up, but I think I got the question.

Code:`If two vectors`

and

have the property that

then there exists a scalar

such that

By definition:

where is the angle between the two vectors.

So the problem statement:

means that

which means that = 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 . (k is positive if the vectors are parallel and negative if they are antiparallel.)

-Dan - November 30th 2006, 07:02 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. - November 30th 2006, 07:16 PMtopsquark
- November 30th 2006, 07:21 PMJenny20
Thus U = k V where k is negative (to switch the direction.)

-Dan[/QUOTE]

Thank you very much.