Results 1 to 7 of 7

Math Help - Vector problem

  1. #1
    Senior Member
    Joined
    Jan 2009
    Posts
    290

    Vector problem

    Given \triangleABC, where points D, E, F are the midpoints of the sides BC, CA, and AB, respectively, and given an arbitrary point P, show that \overrightarrow {PD}+\overrightarrow {PE}+\overrightarrow {PF}=\overrightarrow {PA}+\overrightarrow {PB}+\overrightarrow {PC}.

    (solution in the book)

    Letting \overrightarrow {a}, \overrightarrow {b}, and \overrightarrow {c} be the position vectors of the vertices A, B, and C, respectively (originating from point P)

    \overrightarrow {PD} = \frac {1}{2}(\overrightarrow {b}+\overrightarrow {c})

    \overrightarrow {PE} = \frac {1}{2}(\overrightarrow {a}+\overrightarrow {c})

    \overrightarrow {PF} = \frac {1}{2}(\overrightarrow {a}+\overrightarrow {b})

    Why? How do they get that?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member malaygoel's Avatar
    Joined
    May 2006
    From
    India
    Posts
    648
    Quote Originally Posted by chengbin View Post

    \overrightarrow {PD} = \frac {1}{2}(\overrightarrow {b}+\overrightarrow {c})


    Why? How do they get that?
    from triangle addition of vectors,

    \overrightarrow {PD} = (\overrightarrow {b}+\overrightarrow {BD})

    \overrightarrow {PD} = (\overrightarrow {c}+\overrightarrow {CD})

    adding above two equations,

    2\overrightarrow {PD} = (\overrightarrow {b}+\overrightarrow {BD}+\overrightarrow {c}+\overrightarrow {CD})

    now,
    \overrightarrow {CD}+\overrightarrow {BD}=\overrightarrow {0}
    since D is midpoint....hence the equation follows
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Jan 2009
    Posts
    290
    maylaygoel, thanks for your explaination.

    It makes sense, but I don't get how you got your explanation. I can't imagine an arbitrary point in my mind, which is the main reason I have so much trouble learning vectors.

    Do you mind explaining this?

    <br /> <br />
\overrightarrow {PD} = (\overrightarrow {b}+\overrightarrow {BD})<br />
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    Quote Originally Posted by chengbin View Post
    maylaygoel, thanks for your explaination.

    It makes sense, but I don't get how you got your explanation. I can't imagine an arbitrary point in my mind, which is the main reason I have so much trouble learning vectors.

    Do you mind explaining this?

    <br /> <br />
\overrightarrow {PD} = (\overrightarrow {b}+\overrightarrow {BD})<br />
    Well since <br /> <br />
\overrightarrow {b}<br />
is the position vector of B from point P it comes <br /> <br />
\overrightarrow {PB} = \overrightarrow {b}<br />

    And then <br /> <br />
\overrightarrow {PD} = \overrightarrow {PB}+\overrightarrow {BD} = \overrightarrow {b}+\overrightarrow {BD}<br />
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Jan 2009
    Posts
    290
    Is a picture representation possible? I really don't understand this.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    May 2009
    Posts
    527
    Here's a picture with triangle ABC, the midpoints D, E & F, showing the last thing running-gag posted:
    \overrightarrow {PD} = \overrightarrow {PB}+\overrightarrow {BD} = \overrightarrow {b}+\overrightarrow {BD}<br />

    Vector \overrightarrow {PB} is in red, and \overrightarrow {PD} is in blue.

    I didn't want to show all of the vectors because it would get cluttered.


    01
    Attached Thumbnails Attached Thumbnails Vector problem-vectorprob.jpg  
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Jan 2009
    Posts
    290
    Thanks so much. I get it now.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. vector problem
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: December 10th 2011, 11:17 AM
  2. 3D vector problem
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 6th 2011, 09:59 AM
  3. Vector problem
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: May 3rd 2011, 04:39 PM
  4. vector problem
    Posted in the Algebra Forum
    Replies: 2
    Last Post: March 9th 2010, 09:34 AM
  5. vector problem sum
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 9th 2009, 09:42 AM

Search Tags


/mathhelpforum @mathhelpforum