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Math Help - Dot product

  1. #1
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    Question Dot product

    Question is in the attachment. Please open the attachment. Thank you.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Jenny20 View Post
    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 
    
    
    
    
    \vec{U} and 
    
    
    
    
    \vec{V} have the property that 
    
    
    
    
    
    \vec{U} \cdot \vec{V} = \left | \vec{U} \right | \left | \vec{V} \right | 
    then there exists a scalar 
    
    
    
    
    k \neq 0 such that 
    
    
    
    
    \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:
    \vec{U} \cdot \vec{V} = \left | \vec{U} \right | \left | \vec{V} \right | cos \left ( \theta_{UV} \right )
    where \theta_{UV} is the angle between the two vectors.

    So the problem statement:
    \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
    cos \left ( \theta_{UV} \right ) = 0

    which means that \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 \vec{U} = k \vec{V}. (k is positive if the vectors are parallel and negative if they are antiparallel.)

    -Dan
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  3. #3
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    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.
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Jenny20 View Post
    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.
    Okay, I had missed that negative sign.

    So there is only one possibility for your vectors: cos(\theta_{UV}) = -1 which means that \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
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  5. #5
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    Thus U = k V where k is negative (to switch the direction.)

    -Dan[/QUOTE]

    Thank you very much.
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