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Math Help - vector triple product

  1. #1
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    vector triple product

    If vector b, c, d are non coplanar vectors then prove that the vector r=( aX b)X( c X d)+( aX c )X( dX b)+( aX d)X( bX c) must be parallel to vector a .
    I try to solve this question by using vector triple product and at last i got
    2[a, b, c] d +2[a, d, b] c+2[a, c,d] b
    Where [a, b, c,] Shows scalar triple product and X means cross product. now what to do next .taking cross product with a it should give 0
    But how?
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  2. #2
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    Re: vector triple product

    Quote Originally Posted by ayushdadhwal View Post
    If vector b, c, d are non coplanar vectors then prove that the vector r=( aX b)X( c X d)+( aX c )X( dX b)+( aX d)X( bX c) must be parallel to vector a .
    I try to solve this question by using vector triple product and at last i got
    2[a, b, c] d +2[a, d, b] c+2[a, c,d] b
    Where [a, b, c,] Shows scalar triple product and X means cross product. now what to do next .taking cross product with a it should give 0
    But how?
    Dear ayushdadhwal,

    \vec{r}=(\vec{a}\times\vec{b})\times(\vec{c}\times  \vec{d})+(\vec{a}\times\vec{c})\times(\vec{d} \times\vec{b})+(\vec{a}\times\vec{d})\times(\vec{b  }\times\vec{c})

    From the vector triple product expansion,

    \Rightarrow\vec{r}=2[\vec{a},\vec{b},\vec{c}].\vec{d} +2[\vec{a},\vec{d},\vec{b}].\vec{c}+2[\vec{a},\vec{c},\vec{d}].\vec{b}

    \Rightarrow\vec{a}\times\vec{r}=2\left\{[\vec{a},\vec{b},\vec{c}].(\vec{a}\times\vec{d}) +[\vec{a},\vec{d},\vec{b}].(\vec{a}\times\vec{c})+[\vec{a},\vec{c},\vec{d}].(\vec{a}\times\vec{b})\right\}-------(1)

    Also it follows that,

    \vec{r}=(\vec{a}\times\vec{b})\times(\vec{c}\times  \vec{d})+(\vec{a}\times\vec{c})\times(\vec{d} \times\vec{b})+(\vec{a}\times\vec{d})\times(\vec{b  }\times\vec{c})

    \vec{a}\times\vec{r}=\vec{a}\times\left((\vec{a} \times \vec{b})\times(\vec{c} \times \vec{d})\right)+\vec{a} \times\left((\vec{a} \times \vec{c}) \times (\vec{d} \times\vec{b})\right)+\vec{a} \times \left((\vec{a} \times \vec{d})\times (\vec{b} \times \vec{c})\right)

    From the vector triple product expansion,

    \Rightarrow\vec{a}\times\vec{r}=[\vec{a},\vec{c},\vec{d}].(\vec{a}\times\vec{b})+[\vec{a},\vec{d},\vec{b}].(\vec{a}\times\vec{c})+[\vec{a},\vec{b},\vec{c}].(\vec{a}\times\vec{d})--------(2)

    By (1) and (2) it is clear that, \vec{a}\times\vec{r}=\vec{0} Therefore \vec{a} is parallel to \vec{r}.
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