Results 1 to 10 of 10

Math Help - Three vectors lie on the same line

  1. #1
    Senior Member
    Joined
    Oct 2008
    Posts
    393

    Three vectors lie on the same line

    Initial point on the origin

    v
    1 = (1, 2, 3), v2 = (2,4,6) and v3 = (3, 6, 0)


    How do we do this?

    I know they are linearly dependent as 2v1 + v2 + 0v3 = 0
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1585
    Awards
    1
    Quote Originally Posted by adam_leeds View Post
    Initial point on the origin

    v
    1 = (1, 2, 3), v2 = (2,4,6) and v3 = (3, 6, 0)


    How do we do this?

    I know they are linearly dependent as 2v1 + v2 + 0v3 = 0
    Two vectors are collinear (parallel) if they are multiples of each other.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Oct 2008
    Posts
    393
    Quote Originally Posted by Plato View Post
    Two vectors are collinear (parallel) if they are multiples of each other.
    So they don't lie on the same line as they are linearly dependent
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,603
    Thanks
    1421
    No, saying that three vectors are linearly dependent means they lie in the same plane so it is still possible that they lie on the same line.

    If they were independent, then they could not be on the same line but knowing that they are dependent doesn't tell you whether they lie on the same line or not. Do as Plato suggested- are they all multiples of one another?
    Last edited by HallsofIvy; August 2nd 2010 at 06:52 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Oct 2008
    Posts
    393
    Quote Originally Posted by HallsofIvy View Post
    No, saying that three vectors are linearly dependent means they lie in the same plane so it is still possible that they lie on the same line.

    If they were [b]independent[b], then they could not be on the same line but knowing that they are dependent doesn't tell you whether they lie on the same line or not. Do as Plato suggested- are they all multiples of one another?
    being multiples of each other is linear dependency though?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,603
    Thanks
    1421
    Yes- but it doean't work the other way: if three vectors are linear dependent, they don't have to be multiples of one another.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Oct 2008
    Posts
    393
    Quote Originally Posted by HallsofIvy View Post
    Yes- but it doean't work the other way: if three vectors are linear dependent, they don't have to be multiples of one another.
    So they lie on the same line as 2v1 + v2 + 0v3 = 0
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member
    Joined
    Oct 2008
    Posts
    393
    Sorry it doesnt, as only 2 of them are multiples of each other.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1585
    Awards
    1
    Quote Originally Posted by adam_leeds View Post
    Initial point on the origin
    v
    1 = (1, 2, 3), v2 = (2,4,6) and v3 = (3, 6, 0)

    How do we do this?
    Letís answer this one and for all
    If the three were collinear then the vectors v_1-v_2~\&~v_3-v_2 would be parallel (multiples of each other).
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,603
    Thanks
    1421
    I hate to add to a "once and for all" but the initial post said that these were vectors with "Initial point on the origin". So it is sufficient to determine whether, say, (2, -4, -6) and (-3, 6, 0) are multiples of (-1, 2, 3).

    Of course (2, -4, -6)= -2(-1, 2, 3) so that is a multiple. But (-3, 6, 0) is NOT a multiple of (-1, 2, 3). -3= 3(-1) and 6= 3(2) but 0 is NOT 3 (3). These vectors are do NOT lie on a single line.

    One more time, adam leeds, showing that three vectors are "dependent", that is showing that 2v1 + v2 + 0v3 = 0, only shows they lie in the same plane, not that they along the same line.

    It is also, of course, true, as Plato says, that v_1- v_2= (-3, 6, 9) and [tex]v_3- v_2= (-5, 10, 6) are not multiples of one another. -5= (5/3)(-3) and 10= (5/3)(6) but 6is NOT equal to (5/3)(9)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Vectors : Image Of LIne
    Posted in the Geometry Forum
    Replies: 2
    Last Post: May 27th 2011, 05:12 AM
  2. Vectors: Cartesian EQN of a line
    Posted in the Geometry Forum
    Replies: 1
    Last Post: May 17th 2011, 08:54 PM
  3. Find two vectors perpendicular to a line
    Posted in the Calculus Forum
    Replies: 3
    Last Post: January 21st 2011, 12:31 PM
  4. Vectors and point on the line segment.. (3d)
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 11th 2008, 04:02 AM
  5. Vectors: Intersection of a line and a plane
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 5th 2008, 08:59 PM

Search Tags


/mathhelpforum @mathhelpforum