Find the components of the vector 𝐴𝐵.
We have the vector 𝐴𝐵 marked on the diagram along with the coordinates of the initial and terminal points. We’re looking for the components of the vector 𝐴𝐵. The components of a vector are written in angle brackets separated by a comma. We’ve decided to call the 𝑥-component of our vector 𝑥 and the 𝑦-component 𝑦. Our task now is to find the values of 𝑥 and 𝑦. These two values give instructions telling you how to get to the terminal point of the vector from the initial point. You go 𝑥 units right and then 𝑦 units up.
Let’s have a look at the diagram again. To get from the initial point 𝐴 to the terminal point 𝐵, we need to go 𝑥 units right and then 𝑦 units up. Just by counting squares, we can see that both 𝑥 and 𝑦 look to be around three units, maybe slightly more. To find the exact values of 𝑥 and 𝑦 and hence the exact components of the vector, we’re going to have to use the coordinates of 𝐴 and 𝐵. The 𝑥-coordinate of the initial point 𝐴 is negative 1.9 and the 𝑥-coordinate of the terminal point 𝐵 is 1.3. The 𝑥-component of our vector is the change in 𝑥-coordinates. In other words, the difference between the 𝑥-coordinate of the terminal point and the 𝑥-coordinate of the initial point, which we find to be 3.2. And just by counting squares, this seems to be about right.
The 𝑦-coordinate of the initial point 𝐴 is negative 4.4. And the 𝑦-coordinate of the terminal point 𝐵 is negative one. Just like when finding the 𝑥-component, when finding the 𝑦-component, we subtract these values. We find that 𝑦 is 3.4. Here we viewed that subtracting negative 1.9 is the same as adding positive 1.9. And subtracting negative 4.4 is the same as adding 4.4.
Having found these values, we substitute them in. And we see that the components of the vector 𝐴𝐵 are then 3.2, 3.4.