Step by step
The question
Question: O A is the vector a and O B is the vector b, and P divides A B in the ratio 1 : 2: find the vector A P.
1 Find a route
We want to go from A to P.
We go along arrows we know: A O, then O P.
So A P is A O plus O P.
AP = AO + OP
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 P is two thirds a plus one third b.
AP = −a + (23a + 13b)
3 Collect the a and b
The a's together and the b's together: one third b minus one third a.
AP = 13b − 13a
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 (2, -1.33): exactly the step from A to P.
So it is right!
a = (0, 4), b = (6, 0): AP = (2, -1.33) = P(2, 2.67) − 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