Given that 𝐮 is the vector three, one and 𝐯 is the vector two, five, find the components of 𝐮 plus 𝐯.
In this question, we’re given two vectors 𝐮 and 𝐯. And we’re given them in terms of their components. We need to find the sum of 𝐮 and 𝐯. Since we’re given 𝐮 and 𝐯 in terms of their components, we’ll add these vectors together component-wise. Recall, we can add two vectors of the same dimensions together by adding the corresponding components together. So the vector 𝐚, 𝐛 added to the vector 𝐜, 𝐝 is equal to the vector 𝐚 plus 𝐜, 𝐛 plus 𝐝. So in our case, the vector 𝐮 plus the vector 𝐯 is the vector three, one added to the vector two, five. And we want to add the corresponding components together, so we need to add the first components of these two vectors together and the second components of these two vectors together.
This gives us the vector three plus two, one plus five. And of course, we can evaluate each of these expressions. We have three plus two is equal to five and one plus five is equal to six. So this simplifies to give us the vector five, six, which is our final answer.
However, this isn’t the only way we can add two vectors together. Remember, we can always add vectors together graphically. The first component of our vector will represent the horizontal change. And the second component of our vector will represent the vertical change. So let’s check our answer graphically. Now we don’t need an origin to our diagram. In fact, all we need is a grid of unit squares. However, we’ll include an origin anyway.
Let’s start by having our vector 𝐮 be at the origin, the horizontal change in 𝐮 is three, and the vertical change is one. So if our vector starts at the origin of our diagram, it must end at the point three, one. So our vector 𝐮 will look like this. Since we want to add vector 𝐮 to vector 𝐯, we must have the initial point of vector 𝐯 is the terminal point of vector 𝐮. So our vector 𝐯 is going to start at the point three, one. We know the horizontal component of 𝐯 is two and the vertical component is five, so it needs the move two right and five up.
So our vector 𝐯 looks like this. And we can see the terminal point of vector 𝐯 is the point five, six. Now because we’ve constructed our two vectors like this, the vector 𝐮 plus the vector 𝐯 will start at the initial point of vector 𝐮 and end at the terminal point of vector 𝐯. So to find the vector 𝐮 plus 𝐯, we need to see the change in its horizontal component and the change in its vertical component.
Horizontally, the vector 𝐮 plus 𝐯 starts at zero and ends at five, so its horizontal component will be five. And vertically, it starts at zero and ends at six, so its vertical component is going to be six. So we also get the vector five, six, which confirms our answer. Therefore, we were able to show if 𝐮 is the vector three, one and 𝐯 is the vector two, five, then the components of the vector 𝐮 plus 𝐯 is going to be five, six.