Results 1 to 5 of 5

Thread: parametric equation

  1. #1
    Member
    Joined
    Nov 2006
    Posts
    123

    Question parametric equation

    Question
    If Lsub1 has parametric equation x=1+3t, y=1+t, z=4-t, and Lsub2 has parametric equation x=7-6t, y=-2t, z=3+2t, then Lsub1 and Lsub2 are parallel.

    The answer is true.

    Please show me why it is true. Thank you very much.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,854
    Thanks
    138
    Quote Originally Posted by Jenny20 View Post
    Question
    If Lsub1 has parametric equation x=1+3t, y=1+t, z=4-t, and Lsub2 has parametric equation x=7-6t, y=-2t, z=3+2t, then Lsub1 and Lsub2 are parallel.

    The answer is true.
    Please show me why it is true. Thank you very much.
    Hello Jenny,

    your equations describe two straight lines in $\displaystyle \mathbb{R}^3$

    $\displaystyle L_1: [x,y,z]=\underbrace{[1,1,4]}_{\text{fixed point}}+t\cdot \underbrace{[3,1,-1]}_{\text{direction}}$

    $\displaystyle L_2: [x,y,z]=\underbrace{[7,0,3]}_{\text{fixed point}}+t\cdot \underbrace{[-6, -2, 2]}_{\text{direction}}$

    By comparison you can see, that

    $\displaystyle [-6, -2, 2]=(-2) \cdot [3,1,-1]$. That means the direction vectors are collinear: They have the same direction but different length.
    Therefore $\displaystyle L_1$ and $\displaystyle L_2$ are at least parallel. To proof if they are actually the same you have to show that the fixed point of $\displaystyle L_1$ belongs to $\displaystyle L_2$ too.

    EB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2006
    Posts
    123
    Thank you very much , earboth!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    12,028
    Thanks
    848
    Hello, Jenny!

    If $\displaystyle L_1$ has parametric equations: .$\displaystyle \begin{Bmatrix}x\:= & 1+3t\\ y\:= & 1+t \\ z\:= & 4-t\end{Bmatrix}$

    and $\displaystyle L_2$ has parametric equations: .$\displaystyle \begin{Bmatrix} x\:= & 7-6t \\ y\:= & -2t \\ z\:= & 3+2t\end{Bmatrix}$

    then $\displaystyle L_1$ and $\displaystyle L_2$ are parallel.

    Two lines are parallel if their direction vectors are parallel.
    . . (They do not have to be collinear.)

    $\displaystyle L_1$ has direction vector: $\displaystyle \vec{u}\:=\:\langle 3,1,\text{-}1\rangle$
    $\displaystyle L_2$ has direction vector: $\displaystyle \vec{v}\:=\:\langle\text{-}6,\text{-}2,2\rangle \:=\:\text{-}2\langle3,1,\text{-}1\rangle$

    Since $\displaystyle \vec{v} = -2\vec{u}\!:\;\;\vec{u} \parallel \vec{v}$ . . . . Q.E.D.

    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Nov 2006
    Posts
    123
    Two lines are parallel if their direction vectors are parallel.
    . . (They do not have to be collinear.)

    L_1 has direction vector: \vec{u}\:=\:\langle 3,1,\text{-}1\rangle
    L_2 has direction vector: \vec{v}\:=\:\langle\text{-}6,\text{-}2,2\rangle \:=\:\text{-}2\langle3,1,\text{-}1\rangle

    Since \vec{v} = -2\vec{u}\!:\;\;\vec{u} \parallel \vec{v} . . . . Q.E.D.


    Hi Soroban,
    Thank you very much!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Parametric equation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Jun 13th 2011, 01:16 AM
  2. Cartesian Equation and Parametric Equation
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Jul 29th 2010, 08:33 PM
  3. [SOLVED] Parametric equation / Cartesian equation
    Posted in the Trigonometry Forum
    Replies: 9
    Last Post: Jul 21st 2010, 11:54 AM
  4. Replies: 2
    Last Post: May 23rd 2010, 10:46 AM
  5. Parametric Equation to Cartesian Equation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Mar 26th 2008, 11:19 AM

Search Tags


/mathhelpforum @mathhelpforum