### Video Transcript

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

In this question, weโre given two vectors ๐ฎ and ๐ฏ. And weโre given them in terms of their components. We need to find the sum of ๐ฎ and ๐ฏ. Since weโre given ๐ฎ and ๐ฏ in terms of their components, weโll 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 ๐, ๐ added to the vector ๐, ๐ is equal to the vector ๐ plus ๐, ๐ plus ๐. So in our case, the vector ๐ฎ plus the vector ๐ฏ 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 isnโt 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 letโs check our answer graphically. Now we donโt need an origin to our diagram. In fact, all we need is a grid of unit squares. However, weโll include an origin anyway.

Letโs start by having our vector ๐ฎ be at the origin, the horizontal change in ๐ฎ 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 ๐ฎ will look like this. Since we want to add vector ๐ฎ to vector ๐ฏ, we must have the initial point of vector ๐ฏ is the terminal point of vector ๐ฎ. So our vector ๐ฏ is going to start at the point three, one. We know the horizontal component of ๐ฏ is two and the vertical component is five, so it needs the move two right and five up.

So our vector ๐ฏ looks like this. And we can see the terminal point of vector ๐ฏ is the point five, six. Now because weโve constructed our two vectors like this, the vector ๐ฎ plus the vector ๐ฏ will start at the initial point of vector ๐ฎ and end at the terminal point of vector ๐ฏ. So to find the vector ๐ฎ plus ๐ฏ, we need to see the change in its horizontal component and the change in its vertical component.

Horizontally, the vector ๐ฎ plus ๐ฏ 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 ๐ฎ is the vector three, one and ๐ฏ is the vector two, five, then the components of the vector ๐ฎ plus ๐ฏ is going to be five, six.