Calculus

Find parametric equations for the line through the point (0,2,2)that is parallel to the plane x+y+z = 2 and perpendicular to the line x=1+t, y=2−t, z=2t.
(Use the parameter t.)


(x(t), y(t), z(t)) =

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asked by Jale
  1. perpendicular to the direction
    1 i - 1 j + 2 k
    Vxi + Vyj + Vzk
    if perpendicular dot product is 0
    Vx - Vy + 2 Vz = 0

    also parallel to plane x+y+z = constant 2
    so normal to the normal to that plane
    1 i + 1 j + 1 k
    Vxi + Vyj + Vzk
    Vx + Vy + Vz = 0
    so we have

    Vx - Vy + 2 Vz = 0
    Vx + Vy + Vz = 0

    let Vx = 1 and solve for Vy and Vz (only direction, not magnitude is needed)
    - Vy + 2 Vz = -1
    + Vy + 1 Vz = -1
    ------------------ add

    3 Vz = -2
    Vz = -2/3

    Vy = -1 +2/3 = -1/3
    so direction is
    1 i -2/3 j - 1/3 k
    or
    3 i - 2 j - 1 k

    so
    line through point is
    (0 , 2, 2 ) + ( 3, -2, -1) t

    check my arithmetic !

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    posted by Damon

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