 In this video we will present the solution to question number 8 from the practice exam number 3 for Math 2270, in which case we are asked to find the angle between the vectors u given as 1, 0, negative 4 and v given as 1, 5 and negative 7 and we're asked to write the angle in degrees. And so the formula that's going to be useful for us here is going to be the law of cosines in the linear algebra variant of that law of cosines, for which we see the following. If we take the dot product of the two vectors u and v, we take, excuse me, we divide that by the length of u times the length of v, this is going to equal cosine of the angle between them. So that's what we're looking for. So we need to compute the dot product of u and v. So if we take the dot product of u and v, we're going to get 1 plus 0 plus 28. We have to take the length of the vector u, so that's going to be 1 plus 0 plus 16 all inside the square root. And then we're also going to take the length of v, so we get 1 plus 25 plus 49 all inside the square root. This is supposed to equal cosine of theta, we're starting to solve for theta here. Simplifying what we can here. So we get 1 plus 28, which is 29. This will sit above the square root of 17. And this will then also sit above the square root of 75. 1 plus 49 is 50 plus 25 is 75. So this is equal to cosine of theta, like so. In which case then to find theta, we're just going to get, we're going to take arc cosine of this. So theta equals arc cosine of the number 29 over the square root of 17 times the square root of 75, like so. And this actually is the correct answer. This is the exact form of the answer for which the instructions here do tell us that you can put this in for the final answer and you would get full credit. This is mostly for if you come to the test and you forgot your calculator, it would be impossible to do this without your calculator. I shouldn't say impossible, but it would not be practical for you to be able to do this in a short amount of time. And also the skill set to do that is not within the scope of this class or any of the previous classes, the type of numerical analysis required. So you can put the exact answer if you don't have a calculator. But with the scientific calculator, arc graphing calculator, this would be easy to put in. Do make sure it's in degree mode for which when you calculate cosine inverse or sometimes called arc cosine of 29 over the square root of 17 and the square root of 75, your calculator then gives the response that the angle theta would be approximately, approximately 35.7 degrees.