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