Let A = (3.0 m, 20° south of east), B = (2.0 m, north), and C = (5.0 m, 70° south of west). c. Find the magnitude and the direction of D = A＋B＋C.

Video transcript

Hey everyone. So this problem is working with vectors. Let's see what they're asking us for given vector R which equals O plus P plus Q. What is the magnitude and direction of that vector? Given that vector O equals 4.2 kilometers at an angle of 30 degrees north of west. Vector P equals 2.5 kilometers in the west direction. And vector Q equals three kilometers 45 degrees north of east. Alright, so the first thing we're gonna do here is draw out what these vectors look like and then break them into their X and Y components. So we have vector oh and that's 30° north of West. So it's gonna be over here in quadrant two 30° and a magnitude of 4.2 km. We have vector P which is just in the west direction. So that's gonna be just here and that's a magnitude of 2.5 kilometers. And vector Q which is 45 degrees north of east, which puts us in this first quadrant here. That's a magnitude of three km and this angle is 45. So we see now we have O P and Q and we're looking for a vector that sums these three vectors together. So the way that we're going to do that is break each of the vectors into the X and Y components. And then some, the X and the Y is to find our X and our Y. So let's take, 01st, so O X is going to be -4. times the cosine of 30 because we're looking at the X component, this is the negative X direction and its adjacent to this 30 degree angle. So we're going to be using the Kassian, We plug that into our calculators and we get negative, 3.64 km and are the Y component of vector O is going to be 4. positive because we're in the positive Y direction times the sine of Let that in and we get 2.1. So we're gonna do that for each of the other two vectors. So P is a little bit simpler because it only has an X component. There is no Y component. We know that the magnitude is 2.5, it's in the negative X direction. So it's just negative 2.5. And then for Q in the X direction, it's positive X, it's the positive X direction. So we've got three Times The co sign of Equals 2.12. And then in the Y direction, we have three times the sine of 45, which also equals 2. and it's in the positive Y direction um Because we're here in quadrant one. So now we're going to some oh X P X and Q X to find our X. Let's do that over here O X plus P X plus Q X. So negative 3. -2. Plus 2. gives us -4. kilometers in the Y direction. We're gonna do the same thing. It's just gonna be O Y plus Q Y. So we've got 2.1 plus 2. and that gives us 4.2, 2 km. So the next step here is to recall that the magnitude of the vector is the square root of the sum of the squares of its X and Y components. So that looks like negative 4.2 kilometers squared plus 4.22 kilometers squared plug that in we did 5.83km. Now, the last step is to determine the direction of the vector and when we are working with are X and Y components, that means that we have the opposite and the adjacent sides of the angle. So we know that we are working with tangent. So the tangent of Fado is going equal ry over our X Or 4.22 over -4.02. So when we take the inverse tangent of that We are left with 46° And that is 46° north of West because we are in the negative X direction and positive Y direction. So when we look at our potential answers that aligns with answer choice. D that's all we have for this one. We'll see you in the next video.