Question Video: Finding the Sum of Two Vectors in Two Different Ways Mathematics

Let ๐ฎ = <3, โˆ’2> and ๐ฏ = <โˆ’9, 5>. What are the components of ๐ฎ + ๐ฏ? What are the components of ๐ฏ + ๐ฎ?

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Video Transcript

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

In this question, weโ€™re given two vectors ๐ฎ and ๐ฏ. We need to determine vector ๐ฎ plus vector ๐ฏ and vector ๐ฏ plus vector ๐ฎ. Letโ€™s start by finding vector ๐ฎ plus vector ๐ฏ. 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 ๐ฎ plus vector ๐ฏ 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, weโ€™ve shown ๐ฎ plus ๐ฏ is equal to the vector negative six, three.

We could now follow the same process to determine vector ๐ฏ plus vector ๐ฎ. 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 weโ€™re 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š and ๐› of the same dimension, vector ๐š plus vector ๐› is equal to vector ๐› plus vector ๐š.

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

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