Video: Finding the Components of the Sum of Two Vectors in Component Form

Given that ๐ฎ = โŸจโˆ’3, โˆ’1โŸฉ, and ๐ฏ = โŸจโˆ’2, 5โŸฉ, find the components of ๐ฎ + ๐ฏ.

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Video Transcript

Given that ๐ฎ is the vector negative three, negative one and ๐ฏ is the vector negative two, five, find the components of ๐ฎ plus ๐ฏ.

In this question, weโ€™re given two vectors ๐ฎ and ๐ฏ and weโ€™re given these in terms of their components. We need to use these to find the sum of these two vectors, ๐ฎ plus ๐ฏ. We have a few different options for doing this. For example, we could sketch our two vectors ๐ฎ and ๐ฏ and then add these together graphically, and this would work and give us the correct answer. However, because weโ€™re given ๐ฎ and ๐ฏ in terms of their components, it will be easier to just add these together component-wise. To do this, letโ€™s start by recalling how we add two vectors together component-wise.

If we want to add the vector ๐š, ๐› to the vector ๐œ, ๐, then we add the first components together to get ๐š plus ๐œ and we add the second components together to get ๐› plus ๐, giving us the vector ๐š plus ๐œ, ๐› plus ๐. So letโ€™s apply this to add the two vectors ๐ฎ and ๐ฏ together. Remember, thatโ€™s the vector negative three, negative one added to the vector negative two, five. We add the first components of these two vectors together to get negative three plus negative two. Then we add the second components together to get negative one plus five. This gives us the vector negative three plus negative two, negative one plus five.

Now, all we need to do is evaluate the expressions in our components. Negative three plus negative two is negative five, and negative one plus five is equal to four. And this gives us our final answer of the vector negative five, four. Therefore, by adding the vector ๐ฎ negative three, negative one to the vector ๐ฏ negative two, five component-wise, we were able to show that ๐ฎ plus ๐ฏ is the vector negative five, four.

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