Results 1 to 4 of 4

Math Help - Vector-

  1. #1
    Junior Member
    Joined
    May 2006
    Posts
    37

    Vector-

    A tunnel is to be evacated through a hill. In order to define position, coordinates (x,y,z) are taken relative to an origin O such that x is the distance east from O, y is the distance north and z is the verticl distance upwards, with one unit equal to 100m. The tunnel starts at point A(2,3,5) and runs in a straight line in the direction i+j-0.5k.

    b) an old tunnel through the hill has equation r=4i+j+2k+x(7i+15j+0k). Show that the point p on the new tunnel where x=7.5 is directly above a point Q in the old tunnel. Find the separation PQ of the tunnels at this point.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,807
    Thanks
    116
    Quote Originally Posted by kingkaisai2
    A tunnel is to be evacated through a hill. In order to define position, coordinates (x,y,z) are taken relative to an origin O such that x is the distance east from O, y is the distance north and z is the verticl distance upwards, with one unit equal to 100m. The tunnel starts at point A(2,3,5) and runs in a straight line in the direction i+j-0.5k.

    b) an old tunnel through the hill has equation r=4i+j+2k+x(7i+15j+0k). Show that the point p on the new tunnel where x=7.5 is directly above a point Q in the old tunnel. Find the separation PQ of the tunnels at this point.
    Hallo, kingkaisai2,

    you want to calculate the shortest distance between 2 straight lines in \mathbb{R}^3, which are not equal nor parallel nor intersecting. (The German word is "windschief", literally translated it means crooked. But I don't believe that this is the appropriate expression in English).

    The equation of a line in 3D is: g:\vec{x}=\vec{s}+r*\vec{u} , where the vector s determines the startingpoint S and the vector u the direction of the line.

    The old tunnel is described by:
    o:\vec{x}=(2, 3, 5)+r*(1, 1, -0.5) and the new tunnel runs on this line:

    n:\vec{x}=(4, 1, 2)+r*(7, 15, 0)

    These lines are not parallel and they don't intersect. So the shortest distance is calculated by:

    d(o, n)=|\frac{(\overrightarrow{s_1}-\overrightarrow{s_2}) * (cross(\overrightarrow{u_1}, \overrightarrow{u_2}))}{|cross(\overrightarrow{u_1  },\overrightarrow{u_2})|}|

    Plug in the values you know:

    d(o, n)=\frac{((2, 3, 5)-(4, 1, 2))*(7.5, -3.5, 8)}{\sqrt{132.5}}\approx 0.17375

    Greetings

    EB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by earboth
    (The German word is "windschief", literally translated it means crooked. But I don't believe that this is the appropriate expression in English).
    In America it is "skew".
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,807
    Thanks
    116
    Quote Originally Posted by ThePerfectHacker
    In America it is "skew".
    Hallo, TPH,

    thanks a lot. (I hope my description of the situation and my solution was understandable at least)

    Greetings

    EB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 8
    Last Post: January 11th 2011, 10:17 AM
  2. Replies: 2
    Last Post: January 27th 2010, 08:23 PM
  3. Replies: 11
    Last Post: December 23rd 2009, 01:30 AM
  4. Replies: 4
    Last Post: November 5th 2009, 04:03 PM
  5. Replies: 2
    Last Post: October 5th 2009, 03:25 PM

Search Tags


/mathhelpforum @mathhelpforum