Results 1 to 3 of 3

Math Help - Distance between point and line (vectors)

  1. #1
    Member
    Joined
    Apr 2010
    Posts
    156

    Distance between point and line (vectors)

    Find the distance from the point A(7,6,10) to the line passing through B(3,7,0) and C(4,9,-2).

    I know that the vector is \overrightarrow{BC} = \left \langle 1,2,-2 \right \rangle and that the equation of the line is thus \mathbf{x} = (3,7,0) + t(1,2,-2) t\epsilon \mathbb{R} but I dont understand how you can find the distance between A and this line?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Unknown008's Avatar
    Joined
    May 2010
    From
    Mauritius
    Posts
    1,260
    What you can try is to find the vector direction of AB. Then, find the angle CBA by using the dot product of the vectors.

    Have a sketch, BC is a line, AB is another line. Draw the angle CBA. Drop a perpendicular line from A to line CB.

    Now, the perpendicular line, AB and BM, where M is the point where the perpendicular meets BC is a triangle.

    Have now a right angled triangle, an angle and you can find the length AB. Use

    sin(M\hat{B}A) = \frac{|AM|}{|AB|}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,969
    Thanks
    1788
    Awards
    1
    Quote Originally Posted by SyNtHeSiS View Post
    Find the distance from the point A(7,6,10) to the line passing through B(3,7,0) and C(4,9,-2).
    Given a line \ell :P + tD and that R\notin \ell then the distance from R to \ell is d(R;\ell ) = \left\| {\overrightarrow {PR}  - \frac{{\overrightarrow {PR}  \cdot D}}{{D \cdot D}}D} \right\| = \dfrac{{\left\| {\overrightarrow {PR}  \times D} \right\|}}{{\left\| D \right\|}}.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: May 16th 2011, 05:57 AM
  2. Distance from a point to a line
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 25th 2009, 04:02 PM
  3. Replies: 1
    Last Post: April 15th 2009, 08:47 AM
  4. Vectors, Distance Between Plane and Point
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 2nd 2008, 01:00 AM
  5. Replies: 3
    Last Post: February 20th 2008, 11:17 AM

Search Tags


/mathhelpforum @mathhelpforum