OA = a, OB = b, M is the midpoint of AB, find BM

Step by step

The question

Question: O A is the vector a and O B is the vector b, and M is the midpoint of A B: find the vector B M.

1 Find a route

We want to go from B to M.

We go along arrows we know: B O, then O M.

So B M is B O plus O M.

2 Write it with a and b

Careful!

Going against the arrow is the opposite vector: it gets a minus.

B O goes against the arrow b: minus b.

O M is a half a plus a half b.

3 Collect the a and b

The a's together and the b's together: a half a minus a half b.

4 Check

Let's check with numbers: a = (0, 4) and b = (6, 0).

Our answer gives (-3, 2): exactly the step from B to M.

So it is right!

Let's look back.

First: find a route.

Then: write it with a and b.

Then: collect the a and b.

Last: check.