# Math Help - vectors

1. ## 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.

2. Hello hooke
Originally Posted by hooke
$\vec{PQ}$ 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

3. Originally Posted by Grandad
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 .

4. Hello hooke
Originally Posted by hooke
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

5. Originally Posted by Grandad
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 .

6. Hello hooke
Originally Posted by hooke
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