Video Transcript
Given that 𝐮 is the vector two, negative three; 𝐯 is the vector negative five, four; and 𝐰 is the vector three, negative one, find the components of 𝐮 plus 𝐯 plus 𝐰.
In this question, we’re given three vectors, the vector 𝐮, the vector 𝐯, and the vector 𝐰. And we’re given these in terms of their components. We need to find the sum of these three vectors. We have a few different options for doing this. For example, we could sketch our vectors 𝐮, 𝐯, and 𝐰 and then add them together graphically. However, because we’re given 𝐮, 𝐯, and 𝐰 in terms of their components, it will be easier to add them together component-wise. So we’ll start by writing our sum out in full. We have 𝐮 plus 𝐯 plus 𝐰 is equal to the vector two, negative three added to the vector negative five, four added to the vector three, negative one.
Now we have a few different options for adding these together, and all of them will give us the same answer. For example, we could add vector 𝐮 to vector 𝐯 and then add the resulting vector to vector 𝐰. Alternatively, we could add vector 𝐯 to vector 𝐰 and then add the resulting vector to vector 𝐮. Or, we can just add all three of our vectors together component-wise. It doesn’t matter which method you would want to use. It’s all personal preference. However, it is worth pointing out the reason we’re allowed to evaluate the addition in either order is because vector addition is associative, and in fact we could prove this. All we would need to do is use the component-wise definition of vector addition and then use the associativity of real numbers.
In this video, we’re going to add all of our vectors together at the same time. To do this, all we need to do is add all of the corresponding components. To find the first component of the sum, we need to add all of the first components of our three vectors together. That’s two plus negative five plus three. And to find the second component of the sum, we need to add all of the second components of our vectors together. That’s negative three plus four plus negative one. Now all we need to do is calculate the two expressions in our components together. We have two plus negative five plus three is equal to zero. And negative three plus four plus negative one is also equal to zero. And this gives us our final answer, which is the vector zero, zero.
Therefore, we were able to show if 𝐮 is the vector two, negative three and 𝐯 is the vector negative five, four and 𝐰 is the vector three, negative one, then the components of the vector 𝐮 plus 𝐯 plus 𝐰 will be zero, zero.
