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 O M.
1 Find a route
We want to go from O to M.
We go along arrows we know: O A, then A M.
So O M is O A plus A M.
OM = OA + AM
2 Write it with a and b
Careful!
A B goes from A to B: b minus a, not a minus b.
Careful: not AM = ½(a − b) ✗
O A is a.
A M is half of A B, and A B is b minus a.
OM = a + (12b − 12a)
3 Collect the a and b
The a's together and the b's together: a half a plus a half b.
OM = 12a + 12b
a terms + a terms, b terms + b terms
4 Check
Let's check with numbers: a = (0, 4) and b = (6, 0).
Our answer gives (3, 2): exactly the step from O to M.
So it is right!
a = (0, 4), b = (6, 0): OM = (3, 2) = M(3, 2) − O(0, 0)
Let's look back.
First: find a route.
Then: write it with a and b.
Then: collect the a and b.
Last: check.
Answer
a terms + a terms, b terms + b terms