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## Precalculus

### Course: Precalculus>Unit 6

Lesson 5: Vector addition and subtraction

# Subtracting vectors with parallelogram rule

The parallelogram rule says that if we place two vectors so they have the same initial point, and then complete the vectors into a parallelogram, then the sum of the vectors is the directed diagonal that starts at the same point as the vectors. To subtract two vectors, we simply add the first vector and the opposite of the second vector, i.e., a+b=a+(-b). Created by Sal Khan.

## Want to join the conversation?

• why can't we subtract vectors directly? why do we have to make an algebraic manipulation(a-b -> a+(-b))? • There is a way to subtract vectors more directly, but sadly this video doesn't really show that.

Imagine that we have three points 𝐴, 𝐵, and 𝐶
such that vector 𝒂 starts at 𝐴 and ends at 𝐵
and vector 𝒃 starts at 𝐴 and ends at 𝐶.

Then, vector −𝒃 starts at 𝐶 and ends at 𝐴.

Thereby, vector −𝒃 + 𝒂 starts at 𝐶 and ends at 𝐵.

So, by placing 𝒂 and 𝒃 so that they have the same starting point,
𝒂 − 𝒃 is the vector that starts at the endpoint of 𝒃 and ends at the endpoint of 𝒂.