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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))?(2 votes)
- 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.(6 votes)

## Video transcript

- [Instructor] In this
video, we're gonna think about what it means to subtract vectors, especially in the context
of what we talked about as the parallelogram rule. So let's say we want
to start with vector a and from that we want
to subtract vector b. And we have vectors a and b depicted here. What do you think this is going to be? What do you think is going
to be the resulting vector? Pause this video and think about that. All right. Now the key thing to realize
is a minus b is the same thing as vector a plus the negative of vector b. Now, what is the negative
of vector b look like? Well, that's going to be a vector that has the exact same
magnitude as vector b but just in the opposite direction. For example this vector right over here would be the vector -b. Now we just have to think about what is vector a plus the vector -b? Well, there's two ways
of thinking about that. I could put the tails of both of them at the same starting point,
might as well do the origin. So let me draw -b over here. So we know the vector -b looks like that. So one way that you are probably familiar is you have vector a and then
what you do is you take a copy or you could think of shifting vector b so its tail starts at
the head of vector a. And if you did that, it
would look like this. It would look like this. This is also the vector -b. And then the sum of vector a and vector -b is going to be going
from the tail of vector a to the head of vector -b. So this would be the
result, right over here. Which you could view
as the sum of a plus -b or the difference of vectors a and b or vector a minus vector b. Now, if we wanna think about it in terms of the parallelogram rule, we could take another copy of vector a and put it so that it's
tail's at the head of this -b and then we would get it right over here and we are forming the parallelogram. And then the resulting vector is the diagonal of the parallelogram. And this just helps us appreciate that we could start with -b and then add vector a to that. Or we could start with vector a and then add -b to that. But either way you get this
white vector right over here which we can view as the
vector a minus vector b, and we're done.