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Calculus

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Find the shortest distance from a point P(2,-1,2) to a line L r= [-1,0,7] + t [4,1,-2].

  • Calculus - ,

    Find the shortest distance from a point P(2,-1,2) to a line L r= [-1,0,7] + t [4,1,-2].


    Assume a point Q on the line such that PQ is the shortest possible distance between them.
    Then PQ is orthogonal to the line L.

    Given P(2,-1,2).
    Let Q(-1+4t, 0+t, 8-2t), then
    PQ=<-3+4t, t-1, 6-2t>
    and we look for the value of t such that
    PQ.<4,1,-2>=0
    or
    <-3+4t, t-1, 6-2t>.<4,1,-2>=0

    4(-3+4t)+(t-1)-2(6-2t)=0
    Can you solve for t?

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