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### Course: Precalculus > Unit 6

Lesson 5: Vector addition and subtraction# Parallelogram rule for vector addition

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. Created by Sal Khan.

## Want to join the conversation?

- I really like Sal's more recent approach to videos. This one is so simple yet so comprehensive.

Thanks Sal. Thanks to the team.(19 votes)- For tips and thanks(1 vote)

- Could you prove addition for integers is commutative using this principle(since adding integers is the same as adding a vector with y component zero to a point in the complex plane)?(2 votes)
- The content make sense for me but I have one question: Is it possible to add the head of vector a and b together, instead of adding a vector's tail to the other's head?(1 vote)
- According to the derivation of this law (that i found in google) it was resultant vector = sqroot of (p^2+q^2+2pqcostheta). Here p,q is are two sides and p is the initial side and q is the terminal side. Sal said p+q=r.

but this is not what derivation says the value for r is entirely different [sqroot of (p^2+q^2+2pqcostheta)].Isnt this law incorrect(1 vote)- Don't confuse a vector with its magnitude. If
**p**and**q**are vectors, and the sum of these two vectors is the vector**r**, then we simply write**p**+*q*=**r**.

If we say that**p**has magnitude p and**q**has magnitude q, then the magnitude of**r**is given by your formula (which is essentially just the law of cosines).(1 vote)

- I didn't really understand it. If 2 small sides of a triangle are always more than the remaining side, then how if ->a + ->b = ->c. Shouldn't their sum be more than ->c(1 vote)
- this method is dumb and confusing(0 votes)

## Video transcript

- [Instructor] So we
have two vectors here, vector A and vector B. And what we're gonna do in this video is think about what it
means to add vectors. So for example, how could we think about what does it mean to take vector A and add to that vector B. And as we'll see, we'll
get another third vector. And there's two ways that we
can think about this visually. One way is to say, all right, if we want start with vector
A and then add vector B to it, what we can do, let me
take a copy of vector B and put its tail right
at the head of vector A. Notice I have not changed the magnitude or the direction of vector B. If I did, I would actually
be changing the vector. And when I do it like that,
this defines a third vector which can be use the sum of a plus B. And the sum is going to
start at the tail of vector A and end at the head of vector B here. So let me draw that. So it would look something like that. And we can call this
right over here, vector C. So we could say A plus
B is equal to vector C. Now we could have also thought about it the other way around. We could have said,
let's start with vector B and then add vector A to that. So I'll start with the tail of vector B and then at the head of vector
B, I'm going to put the tail of vector A. So it could look something like that. And then once again, the sum
is going to have its tail at our starting point here and its head at our finishing point. Now, another way of thinking about it is we've just constructed a parallelogram with these two vectors by putting both of their tails together,
and then by taking a copy of each of them and
putting that copy's tail at the head of the other vector, you construct a parallelogram like this, and then the sum is
going to be the diagonal of the parallelogram. But hopefully you appreciate
this is the same exact idea. If you just add by
putting the head to tail of the two vectors and
you construct a triangle, the parallelogram just helps us appreciate that you can start with the yellow vector and then the blue vector
or the blue vector first and then the yellow vector. But either way, the sum is
going to be this vector C.