Draw each of the following vectors, label an angle that specifies the vector's direction, and then find the vector's magnitude and direction. (a) A = 3.0i + 7.0j

Video transcript

Welcome back everyone. We are tasked with sketching this vector P right here and finding the magnitude and direction of the vector. So to start out, I'm going to draw an a, a graph here or an axis, a set of axis axis if you will. So we are going to have the negative X axis and the positive Y axis. And we'll see why. In just a second, the first component of our vector is going to be the X component. So what this tells us is that our vector acts in the negative X direction about five increments. And similarly for the Y component in the positive Y direction for four components, what this gives us is the point negative 54 and attaching this to the origin gives us our vector P. So now what we are left with this is what our vector looks like, but we still need to find what the angle measure is with the negative X axis as well as what is the magnitude of P. So let's go ahead and start out with the magnitude of P here. So the magnitude of P is going to be given by the square root of our X component squared plus our Y component squared. What this gives us is the square root of five squared, which is 25 plus four squared, which is 16, giving us 6.4 for the magnitude of P wonderful. So now let's go ahead and find our angle measure. We can say that the tangent of our angle theta is equal to the Y component over the X component. And what I will do to isolate the theta variable is take the inverse tangent of both sides. What this gives us is that theta is equal to the inverse tangent of P Y over P X. Let's go ahead and plug in some values here. This is the same as the inverse tangent of four divided by negative five which when you plug into our calculator, you get negative 38.7 degrees or 38.7 above the negative X axis, which is what exactly we have here. This angle measure is 38.7 degrees and the magnitude of our vector is 6.4 and there gives us our final answer. Thank you all so much for watching. I hope this video helped. We will see you all in the next one.