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Parallel Lines in space

  1. May 16, 2015 #1
    1. The problem statement, all variables and given/known data
    How are the two lines
    r = i + 2j + t(i - k), and r = k + s(-i + k)
    parallel?
    t,s∈ℝ

    2. Relevant equations
    parametric vector equation of a line
    [tex]r-r_0=tv[/tex]
    3. The attempt at a solution
    Tried to find the conditions for lines to be parallel in ℝ^3.
     
  2. jcsd
  3. May 16, 2015 #2

    haruspex

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    Suppose you have the equation as a constant vector ##\vec a## plus some parameter mutiplied by a second constant vector ##\vec b##. What scalar and vector operations can you do to it that would produce parallel lines?
     
  4. May 16, 2015 #3

    HallsofIvy

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    A line in space can be written as [itex]\vec{r}= \vec{r_0}+ \vec{D}t[/itex] where [itex]\vec{r_0}[/itex] is the "position vector" of a single point on the line (the point where t= 0) and [itex]\vec{D}[/itex] is the "direction vector" pointing in the direction of the line. Two lines are parallel if and only if one direction vector is a multiple of the other.

    Edit: Some text removed by a mentor.
     
    Last edited by a moderator: May 16, 2015
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