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Math Help - Vector proof

  1. #1
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    Vector proof

    Given that three vectors \vec a, \vec b, \vec c, are perpendicular to each other (where \vec a, \vec b, \vec c are not \vec 0 and given that p\vec a+q\vec b+r\vec c=\vec 0, show that p=q=r=0.

    I get the solution, but I don't understand something written in my solution book.

    It says

    Since p\vec a+q\vec b+r\vec c=\vec 0

    \vec a(p\vec a+q\vec b+r\vec c)=0

    Notice that in the second line it is just 0, not \vec 0

    Why?
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  2. #2
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    Quote Originally Posted by chengbin View Post
    Given that three vectors \vec a, \vec b, \vec c, are perpendicular to each other (where \vec a, \vec b, \vec c are not \vec 0 and given that p\vec a+q\vec b+r\vec c=\vec 0, show that p=q=r=0.

    I get the solution, but I don't understand something written in my solution book.

    It says

    Since p\vec a+q\vec b+r\vec c=\vec 0 <<<<< the result is the null vector which is perpendicular to any other vector

    \vec a(p\vec a+q\vec b+r\vec c)=0 <<<<<< since the null vector is perpendicular to any other vector the scalar (dot) product must be zero. (A dot product yields a real number not a vector!)

    Notice that in the second line it is just 0, not \vec 0

    Why?
    ...
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  3. #3
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    Scalar Product

    Hello chengbin
    Quote Originally Posted by chengbin View Post
    Given that three vectors \vec a, \vec b, \vec c, are perpendicular to each other (where \vec a, \vec b, \vec c are not \vec 0 and given that p\vec a+q\vec b+r\vec c=\vec 0, show that p=q=r=0.

    I get the solution, but I don't understand something written in my solution book.

    It says

    Since p\vec a+q\vec b+r\vec c=\vec 0

    \vec a(p\vec a+q\vec b+r\vec c)=0

    Notice that in the second line it is just 0, not \vec 0

    Why?
    \vec a(p\vec a+q\vec b+r\vec c) is the scalar or dot product of the vector \vec a with the vector p\vec a+q\vec b+r\vec c, and is more correctly written \vec a.(p\vec a+q\vec b+r\vec c)

    Since the vector (p\vec a+q\vec b+r\vec c) = \vec 0, this scalar product is the scalar 0.

    The proof then (presumably) continues by noting that \vec a.\vec b = 0 = \vec a . \vec c, since the scalar product of perpendicular vectors is zero.

    Hence p \vec a.\vec a = 0, and hence (if \vec a is a non-zero vector) p = 0.

    Similarly q = r = 0.

    Grandad
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