Video Transcript
Let ๐ฎ be the vector three, negative two and ๐ฏ be the vector negative nine, five. What are the components of ๐ฎ plus ๐ฏ? What are the components of ๐ฏ plus ๐ฎ?
In this question, weโre given two vectors ๐ฎ and ๐ฏ, and we need to find two things. First, we need to find the components of ๐ฎ plus ๐ฏ, and then we need to find the components of ๐ฏ plus ๐ฎ. And both of these involve adding two vectors together, so letโs recall how we do this. In fact, we know several different ways of adding two vectors together. For example, we know how to do this graphically. For example, if we sketch the vector ๐ and then at the terminal point of vector ๐ we sketched our vector ๐, then the vector ๐ plus the vector ๐ will be the vector that follows these two vectors ๐ and ๐. However, this isnโt the only way of calculating the sum of two vectors. We can also do this by using their components.
We recall we can add two vectors of the same dimension by just adding their components together. In other words, the vector ๐ฎ ๐ฅ, ๐ฎ ๐ฆ plus the vector ๐ฏ ๐ฅ, ๐ฏ ๐ฆ will be equal to the vector ๐ฎ ๐ฅ plus ๐ฏ ๐ฅ, ๐ฎ ๐ฆ plus ๐ฏ ๐ฆ. And in this case, this method is going to be easier because weโre already given our vectors in component form. So letโs use this to add our two vectors ๐ฎ and ๐ฏ together. We need to add the vector three, negative two to the vector negative nine, five. All we do is add the first components of each of our two vectors together and then add the second components of our two vectors together. This gives us the vector three plus negative nine, negative two plus five. And we can calculate these. Three plus negative nine is equal to negative six and negative two plus five is equal to three. So this gives us the vector negative six, three.
We can then do exactly the same to find the vector ๐ฏ plus the vector ๐ฎ. This time, we need to add the vector negative nine, five to the vector three, negative two. Once again, we just add the components of these two vectors together. This gives us the vector negative nine plus three, five plus negative two. And we can calculate both of these. Negative nine plus three is negative six, and five plus negative two is three. So this gives us the vector negative six, three. And we could stop here. However, we can notice something interesting. Weโve just shown the vector ๐ฎ plus the vector ๐ฏ is equal to the vector ๐ฏ plus the vector ๐ฎ. And thereโs a few different reasons to see why this is true.
One way is to notice that addition of real numbers is commutative. So when weโre adding two vectors in a different order, all weโre doing is adding the same components together. However, weโre adding these in a different order. And we can just switch this order around, so it doesnโt matter which order we add our two vectors. Therefore, what weโve shown is for any two vectors which we can add together, vector ๐ plus vector ๐ will always be equal to vector ๐ plus vector ๐. And itโs worth pointing out we could also think about this graphically.
If in our same diagram we wanted to add the vector ๐ to the vector ๐, we could instead start with the vector ๐. And then at the terminal point of this vector ๐, we can add our vector ๐. Then, vector ๐ is parallel with vector ๐ and vector ๐ is parallel with vector ๐, and they both start and end at the same point. In other words, weโve just made a parallelogram. And the diagonal of this parallelogram will be equal to both the vector ๐ plus the vector ๐ and the vector ๐ plus the vector ๐. In other words, these two are equal.
Therefore, vector addition is commutative. So we were able to show if ๐ฎ is the vector three, negative two and ๐ฏ is the vector negative nine, five, then the components of ๐ฎ plus ๐ฏ will be negative six, three and the vectors of ๐ฏ plus ๐ฎ will be equal to this. It will also be the vector negative six, three.
