Let's see if we can solve this example. So in this example, we're told if vector a equals 31, and vector b equals negative 49, and vector c is equal to 5 times vector a minus 2 times vector b, we're asked to calculate the magnitude of vector c. Now, what I can see here is that we're trying to ultimately find the magnitude of this vector, but this vector depends on the other two vectors we have in this problem. So we definitely have our work cut out for us here. But whenever solving these problems where we have multiple vector operations, I personally like to start small with the problem and then work my way to solving the whole thing. So let's actually do that.

Now the first thing I'm going to do is figure out what 5 times vector a is, because I'm going really be focusing on this vector c here. And to find 5 times a, well 5 times vector a is just going to be 5 times this vector. So we're going to have 5 times 31. Now, 5 times 31 we can multiply this scalar into each of the components, so we'll have 5 times 31. This is the vector 5a. Next, I'm going to figure out what vector 2b is. So vector 2b is going to be 2 times vector b. Well, I can see here that vector b is negative 49, so that's going to be our vector, and then what I need to do is distribute the scalar into this vector as well. So we're going to have 2 times negative 49. This is vector 2b.

What I need to do from here, they need to subtract these 2 vectors. To find vector c, it's going to be 5a minus 2b. So we can see here that 5a is 155, and then we can see that vector 2b is negative 98. And then what I can do is subtract these 2 vectors. If I subtract them we're going to have 155 minus negative 98, and 155 minus negative 98 is the same thing as 155 + 98, which is 253. So this right here is vector c, 253. And now that I've found vector c, our last step is to just find the magnitude of c.

We can use the Pythagorean theorem, and the Pythagorean theorem for these vectors is going to be the square root of the x component of c squared. Now, 253 squared turns out to be 64,009. So the magnitude of C is the square root of 64,009, and that is the solution to this problem. So this is how you can handle multiple vector operations, as well as finding the magnitude of a vector once you've already done operations on that vector. I hope you found this video helpful. Thanks for watching.