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Find a linear function whose graph is the plane that intersects the xy-plane along the line y=2x+2 and contains the point (1,2,2).

  • Calculus -

    a little vector analysis makes this relatively easy. Find two points B and C on the line. Let A=(1,2,2)

    Form vectors AB and AC. These two vectors lie in the desired plane. Let n be the normal to the plane.

    n = AB × AC
    n•(v-A) = 0 is the desired plane.

    So, pick any two points on the line, say B=(0,2,0) and C=(1,4,0)

    AB = -i -2k
    AC = 2j -2k

    n =
    | i j k |
    |-1 0 -2 |
    | 0 2 -2 |
    = 4i -2j + 2k

    4i -2j + 2k • (x-1)i + (y-2)j + (z-2)k = 0
    2x-y+z = 2

    For a fuller explanation, see

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