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Math Help - Equations of Planes 5

  1. #1
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    Equations of Planes 5

    Find the scalar equation of the plane containing the parallel lines:
    r=(2,2,-1)+s(3,-3,1) and r=(1,2,-3)+t(3,-3,1)

    Can someone check my work for accurateness? I don't have the answer to this question and would like to know if it is correct. Thanks for your help

    P(2,2,-1)
    Q(1,2,-3)
    d3=PQ=(-1,0,-2)
    n=d1 x d3
    n=(6,5,-3)
    6x+5y-3z+D=0
    6(1)+5(2)-3(-3)+D=0
    D=-25

    Final answer:
    6x+5y-3z-25=0
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  2. #2
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    Hello, skeske1234!

    Find the scalar equation of the plane containing the parallel lines:
    . . r\:=\:(2,2,-1)+s(3,-3,1)\:\text{ and }\:r\:=\:(1,2,-3)+t(3,-3,1)

    Can someone check my work for accurateness?
    I don't have the answer to this question and would like to know if it is correct.

    P(2,2,-1),\;Q(1,2,-3)

    d_3\:=\:PQ\:=\:(-1,0,-2)

    n\:=\:d_1 \times d_3 \:=\:(6,5,-3)

    6x+5y-3z+D\:=\:0
    6(1)+5(2)-3(-3)+D\:=\:0 \quad\Rightarrow\quad D\:=\:-25

    Final answer: . 6x+5y-3z-25\:=\:0
    Correct!


    You can check your answer.
    Are the two lines contained in your plane?

    The lines are: . L_1\;\begin{Bmatrix}x &=& 2 + 3s \\ y &=& 2 - 3s \\ z &=& \text{-}1 + s \end{Bmatrix} \qquad L_2\;\begin{Bmatrix}x &=& 1 + 3t \\ y &=& 2 - 3t \\ z &=& \text{-}3 + t \end{Bmatrix}


    Is L_1 in your plane: . 6x + 5y - 3z - 25 \:=\:0 ?

    . . \begin{array}{cccc}6(2+3s) + 5(2-3s) - 3(\text{-}1 + s) - 25 \:=\:0 & & \text{Is this true?} \\<br /> <br />
12 + 18s + 10 - 15s + 3 - 3s - 25 \:=\:0 \\<br /> <br />
0 \:=\:0 & & \text{Yes!}\end{array}



    Is L_2 in your plane: . 6x + 5y - 3z - 25 \:=\:0 ?

    . . \begin{array}{cccc}6(1+3t) + 5(2-3t) - 3(\text{-}3 + t) - 25 \:=\:0 & & \text{Is this true?} \\<br /> <br />
6 + 18t + 10 - 15t + 9 - 3t - 25 \:=\:0 \\<br /> <br />
0 \:=\:0 & & \text{Yes!}\end{array}


    Both lines are contained in your plane.

    Therefore, your equation is correct.

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