Question Video: Adding Two Vectors Component-Wise Mathematics • First Year of Secondary School

Video Transcript

Given that 𝚨 is the vector nine, negative 12 and 𝚩 is the vector 13, negative one, find the vector 𝚨 plus the vector 𝚩.

In this question, we have two vectors, and these are given component-wise: the vector 𝚨 and the vector 𝚩. We need to find the sum of these two vectors. And to do this, we have a few options. The easiest way is to recall exactly what we mean by vector addition. If we’re given two vectors of the same dimension and we’re asked to add these together, then we can do this component-wise. For example, the vector 𝑢 𝑥, 𝑢 𝑦 plus the vector 𝑣 𝑥, 𝑣 𝑦 is equal to the vector 𝑢 𝑥 plus 𝑣 𝑥, 𝑢 𝑦 plus 𝑣 𝑦. All we do is add the corresponding components together to create a new vector.

Since in this question the vector 𝚨 and the vector 𝚩 are both two-dimensional vectors and we know this because each vector has two components, we can add them together by using this method. All we need to do is add the corresponding components together. We add the first components together to get nine plus 13. And we add the second components together to get negative 12 plus negative one. Then we just evaluate each of these expressions. Nine plus 13 is equal to 22, and negative 12 plus negative one is equal to negative 13. And this gives us our final answer of the vector 22, negative 13.

However, this isn’t the only way of answering this question, and we can do this graphically. However, it will take a little bit more work. To do this, we first need to recall when we’re given a two-dimensional vector, we can sketch this by having the first component be the horizontal component of our vector and the second component be the vertical component of our vector. So in our case, the vector 𝚨 will have horizontal component nine and vertical component negative 12. So for our vector 𝚨, the change in 𝑥 needs to be nine, and the change in 𝑦 needs to be negative 12.

And one way of visualizing this is that we start at the initial point of our vector 𝚨 and end at the terminal point of our vector 𝚨. Then we needed to increase our horizontal coordinate by nine and decrease our vertical coordinate by 12. We’re then going to want to do exactly the same for our vector 𝚩. However, we need to notice something first. We want to calculate the vector 𝚨 plus the vector 𝚩. Since when we add vectors together, we travel along both of these vectors, our vector 𝚩 is going to need to start at the terminal point of vector 𝚨. Keeping that in mind, we’re going to need the vector 𝚩 to have horizontal component 13 and vertical component negative one.

And we’re going to start at the terminal point of vector 𝚨. This gives us the following sketch for vector 𝚩. Vector 𝚩 has horizontal component 13 and vertical component negative one. We’re now ready to find the vector 𝚨 plus the vector 𝚩. And it’s easy to do this graphically now because we’ve set our vectors 𝚨 and 𝚩 up in this manner. Now the vector 𝚨 plus 𝚩 is going to start at the initial point of vector 𝚨 and end at the terminal point of vector 𝚩. And we can see, in our diagram, the horizontal component of vector 𝚨 plus vector 𝚩 is just going to be nine plus 13. And the vertical component is going to be negative 12 plus negative one, giving us the exact same answer.

Therefore, we were able to show if 𝚨 is the vector nine, negative 12 and 𝚩 is the vector 13, negative one, then the vector 𝚨 plus the vector 𝚩 will be equal to the vector 22, negative 13.