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Math Help - Vector equation of a line problem

  1. #1
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    Vector equation of a line problem

    I need to find a line that passes through the point P (4,3,2) and is perpendicular to the line (1+2t, -1+t, 1-3t).

    I only managed to use the dot product to get 2x +y-3Z =0 I have no idea how to proceed to get the exact values of x,y,z that passes through 4,3,2.

    Any help will be appreciated.
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  2. #2
    MHF Contributor
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    Hi

    The point P (4,3,2) and the line (1+2t, -1+t, 1-3t) determine a plane.
    You can draw a sketch of this plane if you like.
    Let H(x_H,y_H,z_H) be the orthogonal projection of P on the line.

    H being on the line there exists t_H such that
    x_H = 1+2t_H
    y_H = -1+t_H
    z_H = 1-3t_H

    (PH) being perpendicular to the line the dot product of PH and (2,1,-3) is equal to 0
    2(x_H-4)+(y_H-3)-3(z_H-2)=0

    Substituting x_H with t_H and so on ... you can fnd the value of t_H then the coordinates of H, then the coordinates of PH, from which you can find an equation of (PH)
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