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Math Help - vector question 3

  1. #1
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    vector question 3

    Show that if vectors a, b, c and d lie in the same plane, then (a b) (c d) = 0.
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  2. #2
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    Hello, daphnewoon!

    I have an "eyeball" proof . . .


    Show that if vectors \vec{a}, \vec{b}, \vec{c},\vec{d} lie in the same plane,

    . . then: . (\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) \:=\:\vec{0}

    \text{Since }\vec{a}\text{ and }\vec{b}\text{ are in the same plane, }\vec{m} \:=\:\vec{a}\times\vec{b}\:\text{ is normal to the plane.}

    \text{Since }\vec{c}\text{ and }\vec{d}\text{ are in the same plane, }\vec{n} \:=\:\vec{c} \times\vec{d}\:\text{ is normal to the plane.}

    \text{Since }\vec{a},\vec{b},\vec{c},\vec{d}\text{ are in the same plane: }\:\vec{m} \parallel \vec{n}


    \text{We have: }\;(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) \;=\;\vec{m}\times\vec{n},\;\text{ where }\vec{m} \parallel \vec{n}

    . . \text{and the cross-product of two parallel vectors is }\vec{0}.


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