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Math Help - T/F about orthogonal

  1. #1
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    Question T/F about orthogonal

    question 1
    The nonzero vector U and V are orthogonal if and only if U x V = 0 .
    The answer is false.

    question 2
    U x V = 0 if and only if U and V are orthogonal.

    Can you explain to me why question 1 is false ?
    I know question 2 is true because it is the definition of orthogonal.

    Thank you very much.
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  2. #2
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    Quote Originally Posted by Jenny20 View Post
    question 1
    The nonzero vector U and V are orthogonal if and only if U x V = 0 .
    The answer is false.

    question 2
    U x V = 0 if and only if U and V are orthogonal.

    Can you explain to me why question 1 is false ?
    I know question 2 is true because it is the definition of orthogonal.

    Thank you very much.
    False, False, False.
    Two vectors are othrogonal if and only if their dot product is zero.
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  3. #3
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    Hello, Jenny!

    1) The nonzero vectors \vec{u} and \vec{v} are orthogonal if and only if \vec{u} \times \vec{v}\:=\:0

    The answer is false . . . Right!

    Two vectors are orthogonal if their dot product is zero: . \vec{u}\cdot\vec{v}\,=\,0



    2) \vec{u} \times \vec{v}\:=\:0 if and only if \vec{u} and \vec{v} are orthogonal.
    .
    Strange . . . This is identical to question 1.

    \vec{u} \times \vec{v}\,=\,0 \;\;\longleftrightarrow\;\;\vec{u} \parallel \vec{v}

    The cross product is zero if and only if the vectors are parallel.

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  4. #4
    Newbie Jay Gatsby's Avatar
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    Question 2 is not identical to Question 1. The assumption that the vectors are nonzero has been dropped.

    It's still false though for essentially the same reasons that Question 1 is false.
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  5. #5
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    \begin{array}{l}<br />
 I: < 1,0,0 > \quad \& \quad J: < 0,1,0 >  \\ <br />
 I \cdot J = 0\quad \& \quad I \times J =  < 0,0,1 >  \\ <br />
 \end{array}
    The above shows that #1 is false.

    \begin{array}{l}<br />
 A: < 2,0,0 > \quad \& \quad B: < 1,0,0 >  \\ <br />
 A \times B =  < 0,0,0 > \quad \& \quad A \cdot B \not= 0 \\ <br />
 \end{array}.
    The above shows that #2 is false.
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  6. #6
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    I see.
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