>> You are going to run a distance one mile north. And then you are going to run a distance 0.4 miles west. And then finally, you're going to run a distance 0.1 miles south. Okay; and let's ask the following question, "How far are you from where you started -- And at what angle?" All right. So how do we attack this problem? What do you guys think? What's the next step we should do for this problem? Yes? >> (studnet speaking) Draw a coordinate plane? >> Draw a coordinate system; I like it. So here's my coordinate system. What should I call these two coordinates? Should I call them X and Y, or should I call them something else? What do you think? >> (student speaking) North, east, south, and west? >> North, east, south, and west. How do you remember where east goes, because, "You should Never Eat Sour Watermelon, right, that's how my kids taught me. Okay; north, east, south, west. So now let's see if we can visualize this thing. I'm going to run one mile north. Okay; we're going to start right there. We're going to go one mile north. That's my first movement. Then I'm going to run 0.4 miles west. So west is this way; it's off to the left. And so we're going to go 0.4 miles, which is not quite half of the arrow that we just drew, so maybe something like that. Okay? And then we're only going to run 0.1 mile south; 0.1 is even smaller than that, of course, and so we draw a smallish little arrow like so. And our movement roughly looks like that, we go north, we go west, we go south. And now we need to find this distance right here. How far are we from where we started? Well, let's label some of these things, okay? If that resulting vector is R, let's call the first one A, the second one B, and the third one C. And now all we have to do is add these things up. So let's make a little room down here, and let's see if we can add up. First off, we probably need to know a little bit about A in terms of which direction we're going. So A is all in the northerly direction. There's nothing in east, there's nothing in west. So when we deal with vectors, one thing that we need to worry about is how those directions are going to add up. Okay? So if I'm going north for one mile, and then I go west for .4 miles, does that change how far north I am? No. The only thing that's going to change how far north I am is this last little bit, C, which is minus .1 miles. So the displacement north is going to be what? It is 1.0 minus 0.1; we get 0.9, and our units are miles. It also looks like that we have ended up to the west. So how far west are we? Well, the displacement west is only that middle step, right? The other ones were straight up and down. And so the displacement west is 0.4 miles. And so now look, you actually know the sides of our triangle for R. You know that this vertical one is 0.9, and you know that the horizontal one is 0.4. And so you can calculate R is 0.9 squared, plus 0.4 squared. Add them up, take the square root. Okay. We will revisit this again with regards to unit vectors. But before we even get to unit vectors, you can just approach it this way, just visualize movements north-south independent from movements east-west. And finally, we ask the question, "What angle?" All right. Well, we can pick any angle we want. Why don't we pick this angle right here? And let's call that angle phi. Okay? And let's see if we can calculate what phi is. In the meantime, somebody punch this into your calculator, and tell me what you get for R. Okay. What did you get for R? >> (student speaking) I got 1.044. >> 1.044. Is that right? >> (student speaking) That's wrong, sorry. >> (studnet apeaking) 0.98; [inaudible]. >> It should be a little bit bigger than .9. Did you get an answer? >> (student speaking) .98. >> .98? That's all right; .98. Somebody else concur? Yes? All right, good. All right; so you're a distance .98. And now if we know that, then we know all the sides of the triangle here. We've got a 0.4 up there, we have a 0.9 right there. Now we have a 0.98. And so now we can calculate that angle phi. And you can, again, pick any of the trig relations you want. Let's pick cosine of phi. Where's the right angle? The right angle is there. So cosine of phi is going to be the adjacent, which is 0.9, divided by the hypotenuse, which is 0.98. And if I take the arc cosine of that, what do I get? Somebody plug it in and tell me. It should be a pretty small angle. I would say in the order of 15 degrees or so. What did you get? >> (student speaking) 23.3? >> How much? >> (studnet speaking) 23.3? >> 23.3; does anybody else have that number? Okay; so 23.3 degrees. Now, that's a perfectly valid angle. But whenever you're dealing with north, east, south, and west, and I just give you an angle of 23.3 degrees, you need to specify with regards to what? So what else do we have to add onto here to make this descriptive? What angle is that 23.3 degrees? Is it north; no. Is it west; no. But it's in that quadrant, right, it's up in this direction. So what would I put here; any thoughts? What should I put after that 23.3 degrees? Yes; what do you think? >> (student speaking) Northwest? >> Northwest? Okay; typically whenever we say "northwest," we always mean 45 degrees, okay? Southeast, that's always at a 45-degree angle. So it's halfway between north and west, or halfway between south and east. Okay? So we're in the northwest quadrant, but we need to be a little more specific. What should we add onto that? Yes. >> (student speaking) Did you say north of northwest? >> North of northwest. Okay; now that's an interesting idea. Let's see what north of northwest would look like. Isn't that a movie, "North By Northwest?" Isn't that a Hitchcock? You guys weren't even close to being born, though. Okay, all right. I've got to improve my repertoire of movies. All right; north of northwest. Northwest would be right there. 23.3 degrees north of northwest would be -- let's start at northwest, and let's rotate by 23 degrees until I get to that angle right there. Okay? That would be 23 degrees north of northwest. But that's not quite what we want, right? That angle is not the same as that angle. What we really want is this angle right here. So how do I describe that angle right there? >> (student speaking) Twenty-three degrees west of north? >> All right; let's see if that's right. She said, "23.3 degrees west of north." Okay; what do those words mean? What those words mean is let's start pointing north. And then let's rotate by 23 degrees until we are pointing west of north by that amount. And that, in fact, looks exactly like our angle here, right? That is the angle we want. So this is how you write it, 23.3 degrees west of north. The "of" always means start that way, and then rotate towards the first word, which in this case is "west." All right; and you're going to see those sorts of things a lot on your homework problem, so it's a good idea to sort it out now. All right; hopefully that one's clear. If not, come see me in office hours. Yes; do you have a question? >> (student speaking) Yes. Aren't we trying to find theta, not phi? >> Well, we can describe this angle however we want. So once we get phi and we write it like this, this is plenty descriptive. If you wanted theta, that, okay, we could write that out. What would theta be in this case? If that whole angle is theta, and this is the little angle phi, what would theta be? Yes. >> (student speaking) You just add 90 to it? >> You add 90, right, we've got to go 90 to get to there. And then we've got to add 23.3 degrees. So it would be 90 plus 23.3 degrees, which is 113.3 degrees. Are we really done? We still have to add some words to it, okay? And if that's our answer, 113.3 degrees, how would I tell somebody what that angle is on a map? What are the words that I should put out for that 113.3 degrees? This is kind of a tricky one. What do you guys think? Well, if we follow the same rule, where should we start? What should the "of" be? Yes. >> (student speaking) East? >> East; of east, okay? And if I am of east, I start going east, and then I rotate in the direction of my angle. Am I rotating towards the north, or am I rotating towards the south? I'm rotating towards the north. Okay? So this is a little confusing, because typically you would never say this, right? You would never say 113.3 degrees north of east; you would usually say this answer right here, 23.3 degrees west of north. But in fact, they're the same thing. Okay; they are telling us the same bit of information. Yes; question. >> (student speaking) Would you never hear, "West of east," like 113 point - >> You would never hear, "West of east," that's correct. Right; you would never hear, "West of east," because that doesn't tell you which way to go. Right; if I said, "Forty-five degrees west of east," is that this, or is that this? >> (student speaking) So would that be negative 45? >> Yes; typically what we say is if it's going this way, it's a positive number, and this way it's a negative number. So plus 45 would be up here, minus 45 would be down there. Okay? But on a lot of the homework problems that you're going to see, they don't talk about X and Y, they talk about north, east, south and west. Okay; and so they will ask you to specify, "Are you west of north, are you south of east? What angle are you specifying?" Okay? All right; hopefully, that's clear. If not, come see me in office hours. Cheers.