### Video Transcript

Let ๐ฎ be equal to the vector three, negative two and ๐ฏ be equal to the vector negative nine, five. What are the components of vector ๐ฎ plus vector ๐ฏ? What are the components of vector ๐ฏ plus vector ๐ฎ?

In this question, weโre given two vectors ๐ฎ and ๐ฏ. We need to determine vector ๐ฎ plus vector ๐ฏ and vector ๐ฏ plus vector ๐ฎ. Letโs start by finding vector ๐ฎ plus vector ๐ฏ. We can find this sum by recalling to add two vectors of the same dimension together, we just need to add the corresponding components together. So we add the first component of each vector together to get three plus negative nine, and we add the second component of each vector together to get negative two plus five.

Therefore, vector ๐ฎ plus vector ๐ฏ is the vector three plus negative nine, negative two plus five. Now we evaluate the expression in each component separately. First, three plus negative nine is equal to negative six. Second, negative two plus five is equal to three. This gives us the vector negative six, three. Therefore, weโve shown ๐ฎ plus ๐ฏ is equal to the vector negative six, three.

We could now follow the same process to determine vector ๐ฏ plus vector ๐ฎ. The only difference is we would switch the two vectors around. And this would work; we would get the correct answer using this method. However, there is a slightly simpler method. We can recall that weโre allowed to add vectors of the same dimension in any order. In other words, we know that vector addition is commutative. This tells us for any vectors ๐ and ๐ of the same dimension, vector ๐ plus vector ๐ is equal to vector ๐ plus vector ๐.

Therefore, since vectors ๐ฎ and ๐ฏ are both two-dimensional vectors, we know vector ๐ฏ plus vector ๐ฎ must be equal to vector ๐ฎ plus vector ๐ฏ. And we already found vector ๐ฎ plus vector ๐ฏ to be equal to the vector negative six, three. Therefore, we were able to show if ๐ฎ is the vector three, negative two and ๐ฏ is the vector negative nine, five, then the vector ๐ฎ plus the vector ๐ฏ is the vector negative six, three and the vector ๐ฏ plus the vector ๐ฎ is also the vector negative six, three.