Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This is a fairly simple thing to do, but what would be the optimal approach to solve part (c) (parts (a) and (b) are done):

In the triangle ABC M is the midpoint on AB. Let OA = $\vec{a}$, OC = $\vec{c}$ and OB = $\vec{b}$.

a) Find the vector OM expressed with $\vec{a}$ and $\vec{b}$.
b) The point P is on CM so that CP = 2PM. Find OP expressed with $\vec{a}, \vec{b}$ and $ \vec{c}$.
c) Let N be the midpoint on AC. The point Q is suppose to be on BN so that BQ = 2QN. Show that P = Q.

enter image description here

Logically, how does one go about doing this?

Would you assume that these points are not equal and try to derive a contradiction? Or would you make that point waypoint for another vector sum and show somehow that by implication they are indeed equal?

I know they are equal. In the head of the task the following definition is given; OA = $\vec{a}$, OC = $\vec{c}$ and OB = $\vec{b}$.

share|improve this question
1  
The problem has been hard to understand! It boils down to a standard result about the medians of a triangle $ABC$. They all meet at a point $P$, and any median is divided by $P$ in the ratio $2:1$. The result goes back to Euclid, and probably earlier. There are nice basic geometry proofs. A vector proof is a frequent exercise in a first linear algebra course. –  André Nicolas Feb 2 '12 at 21:53

1 Answer 1

There's no need to try to be smart here.

First do parts (a) and (b) of the exercise. Then do them again with points $B$ and $C$ swapped, which gives you an expression for $OQ$. You will find that the expressions for $OP$ and $OQ$ are identical; hence $P$ and $Q$ must be the same point.

share|improve this answer
    
This is what I get for getting a bachelor in computer science before redoing old high school exams. Maybe I'm reading to much into it. Thanks. –  Algific Feb 2 '12 at 22:35

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.