Question Video: Finding the Sum of Three Vectors in Component Form Mathematics • 12th Grade

Given that 𝐮 = ⟨2, −3⟩, 𝐯 = ⟨3, 2⟩, and 𝐰 = ⟨−1, −5⟩, find the components of 𝐮 + 𝐯 + 𝐰.

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Video Transcript

Given that 𝐮 is the vector two, negative three and 𝐯 is the vector three, two and 𝐰 is the vector negative one, negative five, find the components of 𝐮 plus 𝐯 plus 𝐰.

In this question, we’re given three vectors 𝐮, 𝐯, and 𝐰 and we’re asked to find the sum of these three vectors. And we know a lot of different ways of adding two vectors together. For example, we could sketch all three of these vectors graphically and then add them together graphically. And this would work; however, we’re given these three vectors in terms of their components, so it will be easier to add these 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 three, two added to the vector negative one, negative five. And we recall to add two vectors of the same dimension together, we just need to add the corresponding components together. So we’ll start by adding our vectors 𝐮 and 𝐯 together. When we add these two vectors together, the first component will be the sum of the first components of our vectors. That’s two plus three. And our second component will be the sum of the second components of our two vectors. That’s negative three plus two.

And don’t forget, we still need to add our vector 𝐰. This gives us the following expression. And we can simplify this expression by calculating the components of our first vector. We have two plus three is equal to five and negative three plus two is equal to negative one, giving us 𝐮 plus 𝐯 plus 𝐰 is the vector five, negative one added to the vector negative one, negative five. But now we can see we still need to add two vectors together. So we’re going to need to do this process again. We add the first components of our vectors together and the second components of our vectors together. This gives us the vector five plus negative one, negative one plus negative five.

And we can calculate each of the expressions in our components. Five plus negative one is four, and negative one plus negative five is negative six. So we’ve shown that 𝐮 plus 𝐯 plus 𝐰 is the vector four, negative six. And we could stop here. However, there is something worth pointing out. When we evaluated this expression, we started by finding the vector 𝐮 plus 𝐯 and then we added the result to the vector 𝐰. This isn’t the only option, however. We could’ve also added vector 𝐯 to vector 𝐰 and then added the result to vector 𝐮.

If we were to do this, we would need to add the components of vector 𝐯 and vector 𝐰 together. This gives us the vector two, negative three added to the vector three plus negative one, two plus negative five, which we can simplify to get the vector two, negative three added to the vector two, negative three. And then we can add these two vectors together component-wise. And if we were to do this, the first component of our vector would be two plus two, which is four, and the second component of our vector would be negative three plus negative three, which is negative six.

So we also get the vector four, negative six. So we’ve just shown it didn’t matter in this case which order we added our vectors 𝐮, 𝐯, and 𝐰 together. In fact, this is true in general for any three vectors. We can always add them together in any order we want. This is called the associativity of vector addition. And this is very similar to another property you might have heard of called commutativity, which means we can switch the order of our vectors around. And it’s very useful to keep these properties in mind because often they can be used to simplify problems involving vectors.

Therefore, in this question, we were able to show if 𝐮 is the vector two, negative three and 𝐯 is the vector three, two and 𝐰 is the vector negative one, negative five, then 𝐮 plus 𝐯 plus 𝐰 is the vector four, negative six.