>> I'm showing this problem because once again there is something that may be confusing here but is consistent with what we've been talking about. And the possible confusion is the notation for the vector V. Let's just find the direction angle of V equals 3i minus 4j. Now I mentioned this algebraic form that we can use in dealing with vectors: in describing vectors. But just understand that this means the same thing as in component form 3 negative 4. You see, it means exactly the same thing. It's just that this is a more algebraic form, maybe more manipulative in the algebraic sense than this might be. Now the -- you know, it's not just the matter. When we use this form it's not just the matter of attaching letters to those two. It kind of looks like it and we can kind of think of it like that, but to me mathematically sound, we have to realize that i and j are not really variables here. We're going to treat them like variables for manipulative purposes, but they're not variables. What they are and if we define them properly, if we're thinking about -- well, let's see. We have a horizontal component here and a vertical component here so gee this is a horizontal component. And this must be something that's like a unit horizontal component, you see? And indeed we could define i to be one-zero, you see? So this would be like a unit horizontal vector. And this would be a unit vertical vector. So j could be defined as zero-one. You see? Just for concise mathematical purposes, we would do that. We don't refer back to this very often but it's important to know that it's there. Okay, now understand what we mean by direction angle. If we're talking about this vector, either in this form or this form, we can describe it here, we can draw a diagram consistent with it and it would be a vector that starts at a particular point and has a horizontal component of 3 and a vertical component of negative 4. It's convenient in analyzing vectors to start at the origin and draw the vector from the origin. But really, we could draw that vector from any spot that we want to. Let's understand that. We're just electing to draw it from here. You know it is the case that vectors have characteristics of magnitude and direction, but that magnitude and direction can be taken from any spot that we want to on the coordinate plane? Alright, now the convenience of using the origin as the initial point of the vector is that it's really consistent with our idea of coordinates and so forth from our past studies. Okay, so we go a distance of 3, horizontal component 3, vertical component negative 4. So the vector then looks like this and we want the direction angle. Now the direction angle: gee. If we're thinking about standard form of direction angle, then we would be talking about a direction angle that's measured like this. You see, we want that angle. Okay, now let's see if we can find that angle. We know from our past study that we have a way of calculating magnitude and direction by using a couple of little relationships that we discussed. And the one for the angle is tangent theta is equal to and there are several ways of looking at the fraction over here. It's the vertical component over the horizontal component. It is - if we want to write it in terms of trig ratios - it's sine over cosine. Well we knew this before. And this is consistent with the idea of that unit notion a few moments ago that if we take the Rs out of that R cosine theta R sine theta, then we have simply sine theta over cosine theta you see left for our vertical and horizontal components. So all of this kind of makes sense and blends together. Everything dovetails together in this whole study. Well at any rate if we're making the calculation right from the information given, we would just say tangent theta is equal to negative 4 over 3. Now in the calculator we want the inverse tangent of negative 4 over 3 and the calculator then tells us that theta is negative 53 point 13 degrees. Well this is a reference angle and it's a negatively measured angle. It's the angle right in here. And what we really want is the angle measured from here. So what we do is to calculate the coterminal angle in a positive sense with this angle - this negative angle - is we just add 360 to it. So we would go -- we would just say theta then is 360 with that negative 53 point 13 degrees. And we find the angle we want to be 306 point 87 degrees.