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Math Help - vectors: are these valid?

  1. #1
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    vectors: are these valid?

    If  \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} does it follow that  \vec{b} = \vec{c} ?

    I said that  \vec{a} \cdot (\vec{b}- \vec{c}) = 0 which means that they are perpendicular which implies that  \vec{b} \neq \vec{c} .


    If  \vec{a} \times \vec{b} = \vec{a} \times \vec{c} does it follow that  \vec{b} = \vec{c} ?

    I said that  \vec{a} \times (\vec{b} - \vec{c}) = 0 which means that they are parallel, and so  \vec{b} \neq \vec{c} .
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by shilz222 View Post
    If  \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} does it follow that  \vec{b} = \vec{c} ?

    I said that  \vec{a} \cdot (\vec{b}- \vec{c}) = 0 which means that they are perpendicular which implies that  \vec{b} \neq \vec{c} .
    actually, this means the vector \vec {a} is perpendicular to the vector \vec{b} - \vec {c}. it does not say anything about the vectors \vec{b} and \vec{c} or how they relate to each other.

    If  \vec{a} \times \vec{b} = \vec{a} \times \vec{c} does it follow that  \vec{b} = \vec{c} ?

    I said that  \vec{a} \times (\vec{b} - \vec{c}) = 0 which means that they are parallel, and so  \vec{b} \neq \vec{c} .
    a similar observation can be made here
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  3. #3
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    Because both are false, find counter examples, such as:
    A = \left\langle {1,1,1} \right\rangle ,\;B = \left\langle {1,0, - 1} \right\rangle ,\;C = \left\langle { - 1,1,0} \right\rangle ,\;\& \;A \cdot B = A \cdot C.
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