OA = a, OB = b, P divides AB in the ratio 2 : 1, find BP

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 2 : 1: find the vector B P.

1 Find a route

We want to go from B to P.

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

So B P is B O plus O P.

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 P is one third a plus two thirds b.

3 Collect the a and b

The a's together and the b's together: one third a minus one third b.

4 Check

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

Our answer gives (-2, 1.33): exactly the step from B to P.

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.