Results 1 to 8 of 8

Math Help - vector calc...point of intersection

  1. #1
    Member
    Joined
    Nov 2008
    From
    MD
    Posts
    165

    Question vector calc...point of intersection

    show that the two lines x=(-1/2 t, 2/3 t +1, t- 3/2) and x= (t- 3/2, 2t+3, 3t+ 9/2) intersect. find an equation for the plane that they determine.

    i have no clue how to do this i keep getting stuck.. the professor's hint was:
    Part (1): show that the given lines intersect. In order to do this, you should rename one of the variables given in the expressions for these lines. The book writes both lines using the same variable, t. In order to find out where these two lines meet, you need to first rename the variable in the second line, say, to s. Then set the two expressions equal which sets the three vector components equal and then solve the system of equations that results. You will find a value for t and a value for s. Plugging these values in to their respective lines produces (or should produce, if your math is correct) the same point -- the point of intersection.

    Part (2) then asks you to find the equation of a plane containing these intersecting lines, which I think you can do without a hint


    anyone want to start me off? i changed the variables in the second equation to s's, but what do i set equal? the entire equations or just the x's, y's, and z's? i'm so lost
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Calculus26's Avatar
    Joined
    Mar 2009
    From
    Florida
    Posts
    1,271
    Before going into the details you may want to see my web site
    about intersecting and skew lines in 3 space --notes and animations.

    Vector Valued Functions

    Now with your problem --you can think of the lines as the trajectories of particles. They may either collide in which case they reach the pt of int at the same time. The paths could cross but the particles reach the pt of int at different times whichj is why you need 2 variables.


    (-1/2 t, 2/3 t +1, t- 3/2)

    (s- 3/2, 2s+3, 3s+ 9/2)

    equating the x coord -1/2t = s- 3/2

    t= -2s+3

    equate the y cooords

    2/3t +1 = 2s +3

    2/3(-2s+3)+1 = 2s + 3

    -4s + 9 = 6s +9

    s= 0

    so on L2
    (s- 3/2, 2s+3, 3s+ 9/2) -> (-3/2,3,9/2)

    since t= -2s+3 it follows when s = 0 t= 3

    On L1
    (-1/2 t, 2/3 t +1, t- 3/2) -> (-3/2,3,9/2) as it should
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Calculus26's Avatar
    Joined
    Mar 2009
    From
    Florida
    Posts
    1,271

    Elaboration

    By the way if s = t the particles collide

    and if for the given values of s and t the z coords don't match you woud have skew lines
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Nov 2008
    From
    MD
    Posts
    165
    thanks i was actually on the right track that time, but when i got s=0 for some reason i thought i was doing it wrong. how do i do the second part? is the point (-3/2, 3, 9/2) and the normal is (-3/2, -4/3, -2)?
    not too sure about the normal..i took the difference of the t coordinates in the given problem
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Nov 2008
    From
    MD
    Posts
    165
    or do i take the cross product to find the normal?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Calculus26's Avatar
    Joined
    Mar 2009
    From
    Florida
    Posts
    1,271
    you can use any point on either line (-3/2, 3, 9/2) is fine

    to find N take the crosspro duct of the 2 direction vectors

    -1/2 i +2/3 j + k and i + 2j +3k
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Nov 2008
    From
    MD
    Posts
    165
    (0, 5/2, -5/3) ?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor Calculus26's Avatar
    Joined
    Mar 2009
    From
    Florida
    Posts
    1,271
    yes

    you can also multiply by 6/5 to eliminate fractions and simplify and use

    (0,3,-2)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Is the intersection of two vector spaces a vector space?
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 10th 2011, 11:55 AM
  2. Replies: 1
    Last Post: April 6th 2011, 06:44 AM
  3. Replies: 1
    Last Post: February 13th 2011, 10:16 PM
  4. [SOLVED] Position vector of point of intersection of line and plane
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: July 1st 2010, 03:14 AM
  5. point of intersection of two vector
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 15th 2008, 02:27 PM

Search Tags


/mathhelpforum @mathhelpforum