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Math Help - Matrices and vectors point-line distance

  1. #1
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    Angry Matrices and vectors point-line distance

    I can't find the shortest distance from the point P=<-5,2,-6> to a point on the line given by l:<x,y,z>=<1t,3t,6t>. Can someone help? please?

    This formula is for R^2 but I need R^3
    Should I use it as ...cz+d=0 and ...b^2+c^2? If so how.
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  2. #2
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    Re: Matrices and vectors point-line distance

    SillyMath

    I understand that the parametric equations of the line given to you are x=1t , y=3t ,z=6t
    therefore the vector v parallel to this line is v=(1t,3t,6t) expressed as function of t.
    Now get one point of the line Q(x,y,z)=Q(1t,3t,6t) and define the vector PQ=(t+5,3t-2,6t+6)

    Now get the dot product of the vector v x PQ =0 and determine the value of t.
    thus you have the value of t and you can find the exact coordinates of the point Q and of course the coordinates of the vector PQ.
    GET THE |PQ| AND THIS IS THE DISTANCE OF THE POINT P FROM THE LINE GIVEN TO YOU.

    MINOAS
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  3. #3
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    Re: Matrices and vectors point-line distance

    Quote Originally Posted by SillyMath View Post
    I can't find the shortest distance from the point P=<-5,2,-6> to a point on the line given by l:<x,y,z>=<1t,3t,6t>. Can someone help? please?

    Given the line \ell:Q+tD and a point P the distance from the point to the line is

    d(\ell;P)=\frac{ |\overrightarrow{PQ}  \times D|}{\|D\|}.
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  4. #4
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    Re: Matrices and vectors point-line distance

    The shortest distance is always measured "perpendicularly".

    The line given is x= t, y= 3t, z= 6t. Any plane perpendicular to the line must be of the form 1(x- x_0)+ 3(y- y_0)+ 6(z- z_0)= 0. Taking (x_0, y_0, z_0)= (-5, 2, -6), the plane perpendicular to the line and containing (-5, 2, -6) is (x+ 5)+ 3(y- 2)+ 6(z+ 6)= x+3y+ 6z+ 35= 0.
    The line x= t, y= 3t, z= 6t crosses that plane when t+ 3(3t)+ 6(6t)+ 35= 46t+ 35= 0.

    Solve that for t and solve for x, y, and z where the given line crosses that plane. The distance from that point to (-5, 2, -6) is the shortest distance from the given line to the given point.
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