Shown on the grid of unit squares are the vectors 𝐮, 𝐯, and 𝐮 plus 𝐯. What are the components of 𝐮? What are the components of 𝐯? What are the components of 𝐮 plus 𝐯?
In this question, we’re given a diagram of three vectors: the vector 𝐮, the vector 𝐯, and the vector 𝐮 plus 𝐯. However, instead of our vectors being given with an axes in the origin, instead we’re just given them on a diagram of unit squares. We need to use this grid of unit squares to determine the components of 𝐮, the components of 𝐯, and the components of 𝐮 plus 𝐯. To answer this question, we’re first going to need to recall exactly what we mean by the components of a vector. Now it’s important to remember that vectors are an object with magnitude and direction. They don’t necessarily represent movement. They can represent a lot of different things. However, we can always represent vectors graphically by choosing a coordinate system.
And it can be very useful to think of them in this way. When represented graphically, the vector 𝑎, 𝑏 can represent movement 𝑎 units horizontally and 𝑏 units vertically. Or, if you prefer, the horizontal change will be 𝑎 and the vertical change will be 𝑏. So what does this mean for the three vectors we’ve shown on our grid of unit squares? Well, first, because each of the squares in our grid has unit length, we can just count the number of squares to determine the horizontal change and vertical change in each of our vectors. We can then use this to find the components of our vector because the components of our vector are the values of 𝑎 and 𝑏, the horizontal change and the vertical change.
So let’s start with our vector 𝐮. We need to determine its horizontal change and its vertical change. On our standard coordinate axes, as we move to the right, we increase our horizontal value, and as we move upward, we increase our vertical value. So we can see when we start at the initial point of our vector 𝐮 and end at the terminal point of our vector 𝐮, we move two units to the right and one unit up. So we’ve increased our horizontal value by two and we’ve increased our vertical value by one. So the horizontal component is two and the vertical component of vector 𝐮 is one. Therefore, 𝐮 is the vector two, one.
We can then do exactly the same for vector 𝐯. This time, when we start at the initial point of vector 𝐯 and end at the terminal point of vector 𝐯, we can see we move down four units and to the left three units. Since we move down four units, we’ve decreased our vertical value by four. In other words, our vertical component will be negative four. And because we’ve moved to the left three units, we’ve decreased our horizontal value by three. So the horizontal component of our vector is negative three. Therefore, 𝐯 is the vector negative three, negative four.
Now, all we need to do is exactly the same for our vector 𝐮 plus 𝐯. Let’s see the change in our horizontal position and vertical position when we move from the initial point of our vector 𝐮 plus 𝐯 to the terminal point in our vector 𝐮 plus 𝐯. We see we move one unit to the left and three units down. So the horizontal change is negative one and the vertical change is negative three. This gives us that the vector 𝐮 plus 𝐯 is the vector negative one, negative three.
And we could stop here; however, there is one thing worth pointing out about our vector 𝐮 plus 𝐯. This isn’t the only way we could’ve found this vector. We could’ve also added our two previous vectors together because, remember, when we add two vectors together, we can do this graphically by drawing one vector after the other so the terminal point of our first vector is the initial point of our second vector just like we’ve done in this picture. Then moving along our two vectors is the same as adding these two vectors together.
However, there’s a second way of adding these two vectors together, where we add their components together. And all this really means is we’re adding the horizontal and vertical components of each vector separately. So let’s use this method to check our answer. The vector 𝐮 plus the vector 𝐯 will be the vector two, one plus the vector negative three, negative four. We need to add the corresponding components together. So let’s start by adding the first two components of our vectors together. So the first component is going to be two plus negative three.
And it’s worth reiterating here what this means is if we travel along vector 𝐮 and vector 𝐯, first we travel along vector 𝐮, so we change the horizontal value by two. And then we travel along vector 𝐯, so we change the value by negative three. We can then do exactly the same thing in the vertical direction or for our second components. We add the second component of each vector together to get one plus negative four. And if we calculate both of these components, we get the vector negative one, negative three, just as we did before.
Therefore, given the sketch of the vectors 𝐮, 𝐯, and 𝐮 plus 𝐯 on a grid of unit squares, we were able to find the components of 𝐮, the components of 𝐯, and the components of 𝐮 plus 𝐯. We showed that 𝐮 was the vector two, one; 𝐯 was the vector negative three, negative four; and 𝐮 plus 𝐯 was the vector negative one, negative three.