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Math Help - points in a plane

  1. #1
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    points in a plane

    determine the constant \beta so that the points A=(0,-8,-3),B=(3,\beta,1),C=(-1,0,0),D=(2,1,1) is in a plane.

    I started to put up two vectors with the points A,B,C:

    \bar{AB}=(3,\beta+8,4)

    \bar{AC}=(-1,8,3)

    now i thought that if i choose \beta so that the two vectors makes a plane, i can get the equation for the plane and proof that the point D is in the plane.

    How do i choose \beta so the vectors makes a plane?


    Thanks!
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  2. #2
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    Quote Originally Posted by mechaniac View Post
    determine the constant \beta so that the points A=(0,-8,-3),B=(3,\beta,1),C=(-1,0,0),D=(2,1,1) is in a plane. How do i choose \beta so the vectors makes a plane?
    Can you solve this \overrightarrow {AB}  \cdot \left( {\overrightarrow {AC}  \times \overrightarrow {AD} } \right) = 0~?
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  3. #3
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    yes but isnt it only in space (R^{3}), (\bar{AC}\times\bar{AD}) works?
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  4. #4
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    your vectors ARE in R^3, yes?
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  5. #5
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    oh sorry got confused by the words. thought they want it in R^2 when they say plane and r^3 when it says space :P
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  6. #6
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    Quote Originally Posted by mechaniac View Post
    determine the constant \beta so that the points A=(0,-8,-3),B=(3,\beta,1),C=(-1,0,0),D=(2,1,1) is in a plane.

    I started to put up two vectors with the points A,B,C:

    \bar{AB}=(3,\beta+8,4)

    \bar{AC}=(-1,8,3)

    now i thought that if i choose \beta so that the two vectors makes a plane, i can get the equation for the plane and proof that the point D is in the plane.

    How do i choose \beta so the vectors makes a plane?


    Thanks!
    Any two (independent) vectors will define a plane. You can write the plane in the form A(x- x_0)+ B(y- y_0)+ C(z- z_0)= D where (x_0, y_0, z_0) is a point in the plane (so any of the points you are given) and <A, B, C> is the cross product of the two vectors in the plane.

    So, you could, in fact, write the equation of the line, with \beta in the coefficients, then put the coordinates of each point and find what value of \beta is required so that each point satisfies the equation of the plane.

    However, simpler is what Plato says- the cross product of two vectors in the plane is perpendicular to the plane and so has 0 dot product with any vector in the plane. That is, you can take the cross product of any two of the vectors, say AC ahd AD and then take the dot product of that with the third, AB. The four points will lie in one plane if and only if that is 0.
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