Calculus Grade 12

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Without solving , determine the points of intersection of the line r= (5,-9,3) +k (1,-12,2] and the plane [x,y,z]= (4,-15,-8) + s[1,-3,1] + t[2,3,1], if any exist.

  • Calculus Grade 12 -

    line r= (5,-9,3) +k (1,-12,2]
    x = 5 + k
    y = -9 - 12k
    z = 3 + 2k

    plane:
    x = 4 + s + t
    y = -15 - 3s + 3t
    z = -8 + s + t

    so
    4 + s+ t = 5 + k
    s + t - k = 1 , #1

    -15 - 3s + 3t = -9 - 12k
    -3s + 3t + 12k = 6 , #2

    -8 + s + t = 3 + 2k
    s + t - 2k = 11 , #3

    #1 - #3:
    k = -10


    sub back into line equation:
    r = (5,-9,3) - 10(1,-12,2)
    = (-5 , 111 , -17)

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