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Thread: vector/plane proof

  1. #1
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    vector/plane proof

    Prove that if $\displaystyle a, b, c$ are not all zero, then the equation: $\displaystyle ax+by+cz+d=0$ represents a plane and $\displaystyle \langle a,b,c\rangle$ is a normal vector to the plane.

    Hint if a $\displaystyle \neq$0, then $\displaystyle ax+by+cz+d=0$ is equivalent to $\displaystyle a(x+\frac{d}{d})+b(y-0)+c(z-0)=0$

    My Calc III teacher loves proofs, but I'm not real good with them. How can I prove this? I'm not even sure how to approach it. Can anyone help? Thanks!
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  2. #2
    A Plied Mathematician
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    Well, the usual equation for a plane looks like this:

    $\displaystyle \mathbf{n}\cdot(\mathbf{x}-\mathbf{x}_{0})=0,$

    where $\displaystyle \mathbf{n}$ is a unit vector normal to the surface, $\displaystyle \mathbf{x}$ is a vector to the arbitrary point on the plane, and $\displaystyle \mathbf{x}_{0}$ is a vector to any particular point on the plane.

    Can you get your equation to look like mine? If so, you're done.
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  3. #3
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    $\displaystyle n \cdot (x-x_{0})=0$
    $\displaystyle n=\langle a, b, c\rangle$
    $\displaystyle \langle a, b, c\rangle \cdot (x-x_{0})=0$
    $\displaystyle \langle a,b,c\rangle\langle x,y,z\rangle-\langle a,b,c\rangle\langle x_{0},y_{0},z_{0}\rangle=0$
    $\displaystyle ax+by+cz-ax_{0}-by_{0}-cz_{0}=0$
    $\displaystyle ax+by+cz+d=0$

    I know that $\displaystyle -ax_{0}-by_{0}-cz_{0}=d$, but is this enough to prove it? Where did you get the usual equation for a plane from? Thanks!
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  4. #4
    MHF Contributor

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    It is simply using the fact that two vectors are perpendicular if and only if their dot product is 0. Let $\displaystyle (x_0, y_0, z_0)$ be any fixed point and $\displaystyle (x, y, z)$ be any other point. Then the vector connecting them is [tex]<x- x_0, y- y_0, z- z_0>[tex] and the "set of all points $\displaystyle (x, y, z)$ such that the vector [tex]<x- x_0, y- y_0, z- z_0>[tex] is perpendicular to $\displaystyle <a, b, c>$ (the tangent plane to that vector containing $\displaystyle (x_0, y_0, z_0)$) must satisfy
    $\displaystyle <a, b, c>\cdot<x- x_0, y-y_0, z-z_0>= 0$.
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