For this question were asked to use Vector A and Vector be given an exercise one point for two. And I wrote those out of the top here. Um, I wrote the I hadn't j had components of both Victor and Victor be for part A. It says find the scaler product of the two vectors and far be wants us to find the magnitude between the two factors. So for part A, the scale of product or also known as the dot product, is gonna be a dot b. So this is equal to the I hacked components of a multiplied by the I had components of be So this is gonna be 4.0 multiplied by 5.0. Plus the J had components of a times the J had components to be So this is gonna be 7.0 times and the negative numbers matter. So this is gonna be times negative 2.0 and it gives us a scaler. That's why they call her scaler product. And a scaler is just a number. Meaning it doesn't have any. I had a J hat values associated with it, so we have 4.0 times 5.0 plus 7.0 times negative, 2.0 or, in other words, 20 minus 14 which comes out to be six course, 6.0 weaken box set in as a solution for a part, B wants us to find the angle between the two. Well, another way of calculating the scaler product not be is also by taking the magnitude of a multiplying it by the magnitude of B and the multiplying that by the coast sine of the angle in between them. We're gonna call that angle theta a B, and that's what we're trying to find a state, A B. But we know that this has to come out to equal 6.0 because that's what we found in part a. So then, if we know the magnitude of A and the magnitude of B, we can solve for the angle fate of a B, get a new page here to do this. So first, let's do the magnitude of a. By definition, the magnitude of A is equal to the sum of the square of both sides. So this is gonna be four squared plus seven squared, so it's gonna be 49 plus 16 all square routed or 65 square root. So Route 65 is the magnitude of a now magnitude of B. This is similar to a do the same thing. This is gonna be five squared, plus negative to square. But a negative squared is a positive. So we can just ignore the negative. The negative sign. So five squared, plus two squared all square root. It comes out to be the square root of 29. Now, if you go back to our previous page, we have nine today times magnitude be times the coastline of data A B is equal to 6.0, right? But now we know magnitude of a magnitude of be so we can solve for data. Maybe we find that data maybe is equal to the inverse co sign, right? We have to take the inverse coastline of both sides to get the angle by itself. So in verse, co sign of the ratio of the scaler product which was 6.0 divided by the magnitude of both, which was square root of 65 times square root of 29 so 6.0 divided by route 65 times, Route 25 or 29. Take the inverse coast side of that ratio and we find out this is equal to 82 0.1 degrees bunkum box set in as our solution to the angle between the two vectors.