Question Video: Finding the Resultant of Two Vectors Using the Parallelogram Method Mathematics

Video Transcript

The figure shows two vectors, 𝐯 and 𝐮, where the magnitude of 𝐯 is equal to five and the magnitude of 𝐮 is equal to seven. Use the parallelogram method to find the magnitude of the resultant of these two vectors. Give your answer to two decimal places.

The parallelogram method is so called because of the shape that it creates, as shown in the figure. This says that if two vectors acting simultaneously at a point can be represented both in magnitude and direction by the adjacent sides of a parallelogram drawn from a point. Then the resultant vector is represented both in magnitude and direction by the diagonal of the parallelogram passing through that point.

So as the diagram shows, the resultant of 𝐯 and 𝐮 — that’s their sum — is given by the diagonal of the parallelogram that’s drawn in pink. But how does that help? Well, we’re given some information about the magnitude of the vectors 𝐮 and 𝐯. Remember, the magnitude is the size or the length of the vector. So we can label 𝐮 as seven units and 𝐯 as five units. We can also add in some missing angles. We know that cointerior angles sum to 180 degrees. And we know that these sides are parallel. It’s a parallelogram. So this angle here is given by 180 minus 125, which is equal to 55 degrees.

Now, in fact, we can label the lengths of two further sides in our parallelogram. We know that opposite sides in a parallelogram are equal in length. So we can label these sides as five and seven. And now we’re going to split our parallelogram into two triangles. We’re trying to find the magnitude of 𝐯 plus 𝐮, so the length of that diagonal line.

Let’s call that length 𝑥 units. We now see we have a non-right-angled triangle, for which we know the lengths of two of its sides and the angle between them. This means we can use the cosine rule to find the missing length 𝑥. The cosine rule says that 𝑎 squared is equal to 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 cos 𝐴. Since the angle we have is 55 degrees, we label this vertex 𝐴. Then the side opposite is lowercase 𝑎. We can label the other two sides in any order. Let’s label five as 𝑏 and seven as 𝑐.

We can substitute everything we know about our triangle into this formula. And we get 𝑥 squared equals five squared plus seven squared minus two times five times seven times cos of 55 degrees. Evaluating this expression on the right-hand side gives us 33.84 and so on. Now, we’ll solve this equation for 𝑥 by taking the square root of both sides. That gives us 5.818 and so on, which, correct to two decimal places, is 5.82. So we found the length of 𝑥 to be 5.82 units, which means the magnitude of the resultant of 𝐮 and 𝐯 is 5.82.