Which three vectors shown in the diagram produce a vector with a magnitude of zero when added together?
We have a diagram here showing five different vectors. For three vectors added together to produce a vector with a magnitude of zero means that if we start at the origin and add three vectors together using the tip-to-tail method, they need to end up back at the origin. So in this example, the sum of 𝐀 plus 𝐁 plus 𝐂 is the zero vector, which has a magnitude of zero.
Another way to think about this is if we write our vectors out in component form — so 𝐀 is equal to 𝑎 sub 𝑥 𝐢 hat plus 𝑎 sub 𝑦 𝐣 hat, 𝐁 is equal to 𝑏 sub 𝑥 𝐢 hat plus 𝑏 sub 𝑦 𝐣 hat, and 𝐂 is equal to 𝑐 sub 𝑥 𝐢 hat and 𝑐 sub 𝑦 𝐣 hat — then 𝑎 sub 𝑥 plus 𝑏 sub 𝑥 plus 𝑐 sub 𝑥 must be equal to zero. And 𝑎 sub 𝑦 plus 𝑏 sub 𝑦 plus 𝑐 sub 𝑦 must also be equal to zero.
If we start by looking at the horizontal components of the five vectors we have to choose from, we can see that we have two with negative horizontal components and three with positive horizontal components. In order for the horizontal components to sum to zero, we must have at least one of each positive and negative. We can’t just sum together the three with positive horizontal components as that would just take us further and further away from the origin in the horizontal direction. Therefore, the three vectors that we choose must include 𝐏 or 𝐐.
So let’s start with vector 𝐏. From the tip of vector 𝐏, we need to go in the positive direction both horizontally and vertically, so vector 𝐑 seems like a good choice. So let’s slide vector 𝐑 down so that its tail touches the tip of vector 𝐏. From here we have a problem because from the tip of our new vector 𝐑, we need our third vector to go straight up vertically with no horizontal component. And there is no vector that matches that description. So it looks like starting from vector 𝐏 was not a good choice.
Let’s try instead starting with vector 𝐐. Starting from the tip of vector 𝐐, we need something that goes in the positive direction horizontally and in the negative direction vertically. So we might try vector 𝐒. However, if we slide vector 𝐒 over so that its tail touches the tip of vector 𝐐, we end up on the horizontal axis, which means we now need something that goes in the positive direction horizontally with no vertical component. And we don’t have a vector that matches that description.
But we do have another choice of a vector with a positive horizontal component and a negative vertical component. And that is vector 𝐓. Let’s slide vector 𝐓 over so that its tail touches the tip of vector 𝐐.
Now we need our third vector to have both positive horizontal and positive vertical components. And there’s only one option, which is vector 𝐑. If we slide vector 𝐑 over so that its tail touches the tip of vector 𝐓, then we have our resultant vector, which points back to the origin.
So the sum of the three vectors 𝐐, 𝐓, and 𝐑 gives us our zero vector. So the three vectors that can be added together to produce a vector with a magnitude of zero are 𝐐, 𝐑, and 𝐓.