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 : 4: find the vector O P.
1 Find a route
We want to go from O to P.
We go along arrows we know: O A, then A P.
So O P is O A plus A P.
OP = OA + AP
2 Write it with a and b
Careful!
A B goes from A to B: b minus a, not a minus b.
Careful: not AP = (a − b) ✗
O A is a.
A P is 1 of the 5 parts of A B, and A B is b minus a.
OP = a + (15b − 15a)
3 Collect the a and b
The a's together and the b's together: four fifths a plus one fifth b.
OP = 45a + 15b
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 (1.2, 3.2): exactly the step from O to P.
So it is right!
a = (0, 4), b = (6, 0): OP = (1.2, 3.2) = P(1.2, 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