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Math Help - Cross Product of Vectors

  1. #1
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    Cross Product of Vectors

    I need help answering this question.
    "Use vectors v=2i+3j-k and z+-3i-j-4k to prove that -(v X z) = z X v. Write a summary and show graphically what this means with respect to the cross product v X z."
    I get 13i + 11j + 7k as the cross product of both -(v X z) and z X v so I understand how to prove this algebraically, but I have no idea how i would write a summary about this or show this graphically.

    Help is greatly appreciated.
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  2. #2
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    Quote Originally Posted by collegemath View Post
    I need help answering this question.
    "Use vectors v=2i+3j-k and z+-3i-j-4k to prove that -(v X z) = z X v. Write a summary and show graphically what this means with respect to the cross product v X z."
    I get 13i + 11j + 7k as the cross product of both -(v X z) and z X v so I understand how to prove this algebraically, but I have no idea how i would write a summary about this or show this graphically.

    Help is greatly appreciated.
    \displaystyle -(v\times z)=(-1)\begin{vmatrix}i&j&k\\2&3&-1\\-3&-1&-4\end{vmatrix}=-[i(-12-1)-j(-8-3)+k(-2+9)]

    \displaystyle (z\times v)=\begin{vmatrix}i&j&k\\-3&-1&-4\\2&3&-1\end{vmatrix}=i(1+12)-j(3+8)+k(-9+2)
    Last edited by dwsmith; February 5th 2011 at 08:03 PM.
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  3. #3
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    so how would i write a summary on this or show this graphically?
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  4. #4
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    Quote Originally Posted by collegemath View Post
    so how would i write a summary on this or show this graphically?
    Draw the vectors and the corresponding normal vector (graphically). Since it says write or show graphically, I would show graphically.
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  5. #5
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    Quote Originally Posted by dwsmith View Post
    \displaystyle -(vxz)=(-1)\begin{vmatrix}i&j&k\\2&3&-1\\-3&-1&-4\end{vmatrix}=-[i(-12-1)-j(-8-3)+k(-2+9)]

    \displaystyle (zxv)=\begin{vmatrix}i&j&k\\-3&-1&-4\\2&3&-1\end{vmatrix}=i(1+12)-j(3+8)+k(-9+2)
    LaTeX

    [tex] z\times v[/tex]

    gives:

     z\times v
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  6. #6
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    which vectors do i draw and what is a correspnoding normal vector?
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  7. #7
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    Quote Originally Posted by collegemath View Post
    which vectors do i draw and what is a correspnoding normal vector?
    Draw vectors v, z, and z x v is the normal vector.

    Then draw -v, -z, and -(v x z) is the normal vector.
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  8. #8
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    looks like i also need to write a summary how do i do that?
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  9. #9
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    Quote Originally Posted by collegemath View Post
    looks like i also need to write a summary how do i do that?
    I would first draw the plane v and z and the plane -v and -z. What do you notice?
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  10. #10
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    so is z X v just a reflection above the x - axis of v X z
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  11. #11
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    Quote Originally Posted by collegemath View Post
    so is z X v just a reflection above the x - axis of v X z
    I am really bad at drawing in 3d. I would say it is a reflection across the line perpendicular to the vector.

    I saw this by looking at a 2d example.

    vector <1,0> <-1,0> this would be a reflection across the y axis which is perpendicular to it.

    So I believe you could say is that -(v x z) is a reflection across the normal vector of (z x v).
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