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Math Help - vectors

  1. #1
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    vectors

    \vec{PQ} and \vec{PR} represent the sides of PQ and PR of a triangle PQR respectively . If S is the midpoint of QR , show that the \vec{PQ}+\vec{PR}=2\vec{PS}

    I managed to prove this .

    Using the method above or otherwise , find the position of O which is located in the triangle PQR where \vec{OP}+\vec{OQ}+\vec{OR}=0

    need help on this part.
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  2. #2
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    Hello hooke
    Quote Originally Posted by hooke View Post
    \vec{PQ} and \vec{PR} represent the sides of PQ and PR of a triangle PQR respectively . If S is the midpoint of QR , show that the \vec{PQ}+\vec{PR}=2\vec{PS}

    I managed to prove this .

    Using the method above or otherwise , find the position of O which is located in the triangle PQR where \vec{OP}+\vec{OQ}+\vec{OR}=0

    need help on this part.
    Here's a hint.

    You've just shown that \vec{SP} = -\tfrac12(\vec{PQ}+\vec{PR}). Now find \vec{TQ} and \vec{UR} in the same way, where T, U are the mid-points of RP and PQ respectively.

    Then add \vec{SP}, \vec{TQ} and \vec{UR} together.

    What happens next?

    Grandad
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  3. #3
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    Quote Originally Posted by Grandad View Post
    Hello hookeHere's a hint.

    You've just shown that \vec{SP} = -\tfrac12(\vec{PQ}+\vec{PR}). Now find \vec{TQ} and \vec{UR} in the same way, where T, U are the mid-points of RP and PQ respectively.

    Then add \vec{SP}, \vec{TQ} and \vec{UR} together.

    What happens next?

    Grandad
    thanks , yeah , i got 0 . It looks to me O is the centroid of the triangle but i am not sure how to describe the position of O .
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  4. #4
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    Hello hooke
    Quote Originally Posted by hooke View Post
    thanks , yeah , i got 0 . It looks to me O is the centroid of the triangle but i am not sure how to describe the position of O .
    Just say it's the centroid - that would be good enough for me!

    Grandad
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  5. #5
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    Quote Originally Posted by Grandad View Post
    Hello hookeJust say it's the centroid - that would be good enough for me!

    Grandad
    thanks , but the book says O divides PS into a ratio of 2:1 , i am not sure how to get that .
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  6. #6
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    Hello hooke
    Quote Originally Posted by hooke View Post
    thanks , but the book says O divides PS into a ratio of 2:1 , i am not sure how to get that .
    It is a well-known fact that the centroid divides each of the medians in the ratio 2:1. For a vector proof, see here.

    Grandad
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