Consider two vectors 𝚨 and 𝚩. 𝚨 equals three 𝐢 hat minus three 𝐣 hat, and 𝚩 equals negative four 𝐢 hat plus nine 𝐣 hat. Calculate 𝚨 plus 𝚩.
This question gives us two vectors in component form, and it asks us to calculate their sum. If we look at our two vectors, we see that they each have an 𝑥-component given by the number in front of the unit vector 𝐢 hat and they each have a 𝑦-component given by the number in front of the unit vector 𝐣 hat. Recall that 𝐢 hat is the unit vector in the 𝑥-direction and 𝐣 hat is the unit vector in the 𝑦-direction. In order to add together two vectors, we need to add together the 𝑥-components and the 𝑦-components of those vectors separately. When we do this, the result that we get for the sum of these two vectors is known as their resultant. So, let’s add together our vectors 𝚨 and 𝚩 from the question.
If we add together the 𝑥-components to get the 𝑥-component of our resultant vector, we have three plus negative four. Then, since this is the 𝑥-component, we multiply it by 𝐢 hat. Then, if we add the 𝑦-components of our vectors to get the 𝑦-component of our resultant, we have negative three plus nine. And this 𝑦-component gets multiplied by 𝐣 hat. The last step is then to evaluate these sums for the 𝑥-component and 𝑦-component of our resultant vector. If we do this for the 𝑥-component, we have three plus negative four, which gives us negative one. And if we do this for the 𝑦-component, we have negative three plus nine, which gives us six. So, we have our answer that the sum of the vectors 𝚨 and 𝚩 is equal to negative one 𝐢 hat plus six 𝐣 hat.