Midpoint of P and Q: Finding 1/3 Distance

[sdoyle]

Homework Statement

Let P and Q be points with corresponding vectors p and q.
a) show that the vector 1/2p+1/2q is the midpoint of the line segment joining P and Q
b)Find a point that is one third of the distance from P to Q

Homework Equations

No idea

The Attempt at a Solution

I'm not really sure.. but if you were to consider P to be one unit vector and Q to be one unit vector then PQ would be 2 unit vectors??

 

[AEM]

Well, this should get you started. Draw x-y coordinates. Label the origin O. Pick a point P and a point Q. Draw the vectors OP and OQ (call them P and Q for short). Use the parallelogram construction to add P and Q. Label the vector P + Q = OM. (Call it M for short). Since the long diagonal of the parallelogram is the sum P+Q, what is the short diagonal of the parallelogram joining the points P and Q (in terms of the vectors P and Q)? I know I'm using the labels P and Q in two different senses here, try to keep them straight.

We know from plane geometry that these two diagonals bisect each other. Call the point of intersection N. Then ON can be written as

ON = P + one half times the vector from P to Q.

Then if you figured out the answer to the question I asked, you can fill in the words and answer appears.

I have a hunch there's a quicker solution, but I don't see it right now.

 

[AEM]

Okay, here's a better approach that can be used to solve the second part of your question as well. Draw your vectors OP and OQ. The vector from P to Q is OP-OQ. Call it PQ. Pick the midpoint of PQ and label it N. Now you want to express ON in two ways:
ON = OP + (1/2)*PQ and ON = OQ - (1/2)*PQ (Why the minus sign?) You can easily combine those two equations to eliminate the PQ vector and you will easily be able to show that the (1/2)*OP + (1/2)*OQ = ON, the vector to the midpoint.

Using the same idea you can find the combination of OP and OQ that will give you the vector to a point one third of the way from P to Q.