Step by step
The question
Question: O A is the vector a and O B is the vector b, and N is the midpoint of O B: find the vector A N.
1 Find a route
We want to go from A to N.
We go along arrows we know: A O, then O N.
So A N is A O plus O N.
AN = AO + ON
2 Write it with a and b
Careful!
Going against the arrow is the opposite vector: it gets a minus.
Careful: not AO = a ✗
A O goes against the arrow a: minus a.
O N is half of O B: half b.
AN = −a + 12b
3 Collect the a and b
The a's together and the b's together: a half b minus a.
AN = 12b − a
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, -4): exactly the step from A to N.
So it is right!
a = (0, 4), b = (6, 0): AN = (3, -4) = N(3, 0) − A(0, 4)
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