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Math Help - Equation of a Plane and Proofs

  1. #1
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    Equation of a Plane and Proofs

    I'm not even remotely close to understanding the first problem I dont understand how I can prove this at all. The 2nd problem I really just need help starting If anyone has any inputs at all it would be extremely helpful

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  2. #2
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    The first one is just a matter of substitution and algebra, for example:

    \hat{c}\cdot\hat{b}=c_1b_1+c_2b_2+c_3b_3

    For the second one, you need to find a normal vector. If you label the points A, B, and C (it doesn't matter which is which):

    \hat{n}=(B-A)\times(C-A)

    where (B-A) is the vector from A to B and (C-A) is the vector from A to C. Then the equation of the plane is:

    \hat{n}\cdot(x-A)=0

    Post again in this thread if you're still having trouble.

    - Hollywood
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  3. #3
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    I still dont understand that first problem where you have to prove the equation
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  4. #4
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    Ok, no problem. You just substitute the coordinate notation (a_1,a_2,a_3) for the vector notation \hat{a}.

    \hat{c}\times(\hat{a}\times\hat{b})=(\hat{c}\cdot\  hat{b})\hat{a}-(\hat{c}\cdot\hat{a})\hat{b}

    (c_1,c_2,c_3)\times((a_1,a_2,a_3)\times(b_1,b_2,b_  3)) = ((c_1,c_2,c_3)\cdot(b_1,b_2,b_3))(a_1,a_2,a_3)-((c_1,c_2,c_3)\cdot(a_1,a_2,a_3))(b_1,b_2,b_3)

    (c_1,c_2,c_3)\times(a_2b_3-a_3b_2,a_3b_1-a_1b_3,a_1b_2-a_2b_1) = (c_1b_1+c_2b_2+c_3b_3)(a_1,a_2,a_3)-(c_1a_1+c_2a_2+c_3a_3)(b_1,b_2,b_3)

    (c_2(a_1b_2-a_2b_1)-c_3(a_3b_1-a_1b_3),c_3(a_2b_3-a_3b_2)-c_1(a_1b_2-a_2b_1), c_1(a_3b_1-a_1b_3)-c_2(a_2b_3-a_3b_2)) = (c_1b_1+c_2b_2+c_3b_3)(a_1,a_2,a_3)-(c_1a_1+c_2a_2+c_3a_3)(b_1,b_2,b_3)

    and I think the rest is pretty clear.

    - Hollywood
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