Chapter one, Section seven Problem. Twenty four For the back There's Ambien. Figure you won twenty four is I have drawn here says you scale drawing to find the magnitude and direction of a the vector sum of a plus B So it begin with that. And so the one thing we learned from the vector section is how do we add these vectors together? So we'LL begin with A When you begin the Arab actors is you first start with the beginning to the end and then you add the component of the bee vector So this could be shifted in any direction and in any translation so we could begin way starting off going down eight of the three, eh? And when we add, we continue on from here with vector V upwards. Fifteen. This isn't exactly drawn to scale the thirty two b's Corrective e. And so the way this summation works is our fine electorate. Here goes from the beginning to the end. This Sarah vector, eh? Plus being so he's in terms of a drying. This gives us an idea of what we kind of have going on, but it doesn't really tell us all the information we want to know, like, what are the angles of this and what is magnitude of this? And so that's where basic trigonometry comes into play. And as you see, we're going to need to use the law of co signs. Yes, we need to pull this out from her old trigonometry training. And when we draw the triangle in quick, is there a basic triangle? And in this case, are our side right here was a fifteen and we know the angle. Here I was thirty. That's what we want to find out is the magnitude of a plus B, which is this one. So in this case, this is going to be C here and from this triangle we can see that a that's going to corresponds to the to the to the you will be to the fifteen and we just plug in everything in the sea for responsive to the thirty degrees that is opposite of C. And so when we play this in, it's C is equal to the square root of This is this is the quantity square. So we want to find sea by itself. That's gonna be Hey, I swear less. Fifteen square minus two times, eh? Times fifteen times. Co Sign of thirty degrees. Remember, is thirty degrees not thirty. Radiance. Your calculator might start off default with radiance. Just make sure to check that. So if you would uplevel listen to our calculator, we find that see is actually a nice number. It's equal to nine here, approximately nine. They're here. And so that's great way now. I found our magnitude for a plus B, but the thing is, we don't know the direction Vectors have a direction and a magnitude and direction. So that's where the next part from trigonometry comes into play the lost science. This Ah, good relationship that can give us all these qualities. And so you try the travel again in a week. You now know that this is about nine. This is a and this is fifteen. So one of the singles thirty degrees and we want to find out the angle course this angle as it would correspond away from the X axis. We got our why, Mrs X. So we just want one of the this angle in particular to find out the direction. And so, based on the law, signs we would then use this quantity. So use these parts of the equation. So we'LL have sign of thirty degrees. This what part wants to do the sea part first it's two nine is equal to sign of this angle which was just called be for now over overnight. That corresponds to this one who fifteen? Okay, the way we solve this, he's using our Andy Dandy calculator once again to solve for being. And we do that by actually using the arc sine, which is the Empress function, to get out the value with this normally would be so I'd be the arc sine of mon players to you. Alison fifteen divided by nine. I am sign of thirty degrees. So the answer for this and being fifty six point four degrees they're here, Marie And so we basically have our magnitude, which he said it was nine. And the direction of this vector from police is this bottom part is no the D six point four and so directors funny in that direction away from this quality. If we wantto say the angle away from here, then it would just be the ninety degrees minus this. Okay, And so that is for a plus b the vectors. April's being for party, for part B. We're going to want to find it for a minus. B said the vector difference. And so first thing we have to find out is or were they realizes this is the same as Vector edition of the negative vector. And so the negative actor beaches corresponds to be corresponds to the negative. In this direction, negative B still has the same magnitude fifteen Nobbs a direction and his thirty degrees away from here. So when we actually look at it in the larger view, go down a for a and then we go fifteen and destruction at an angle thirty away from here. And since lines are one hundred eighty degrees, that means that this angle inside is one hundred fifty. The police and so travel looked like this. See, See, See, this is a longer section. And so based on what we had somebody draw the triangle right here. So we know this angle is one hundred fifty fifteen. This is eight. So we want to find this side, see right here. So we do the same thing we've literally used this exact formula or it's a square less fifteen square minus, two times a day times fifteen times. Co sign of one hundred fifty degrees. Just in case you're not familiar with this location. If they put a thought here that means times till we get for here is we get about twenty two, twenty two point three, Percy. So that gives us the length. And I only want to calculate the signs. We do the same thing with the same process of the law. Of course, I So we have twenty two point three here and this angle's one hundred fifty. And we want to find out kissing, which we'LL call V again and what people does that correspond to fifteen. Until we do the same thing with the Ark co sign for the arc sine, we find out that B is equal to nineteen. Wait six, six degrees, and so this angles nineteen degrees. And so this would be our rector a minus b. Okay, so the next tests were given, it says, use the answers to find the magnitude and direction of minus a minus. B. First thing to realize is this's there's something really nice about this. There's the minuses here, which means you could pull a minor self. And so what we get is negative. A plus B and we had calculator was actor like this last time, so it's literally just the opposite. So the man into whose is the same? It's about nine and the the angle we calculator was fifty up fifty six year, which means this is about it's fifty six above here or the opposite. Like we said, ninety minus fifty six. From here, this angle is fifty six, and so the same thing with the other one word says B minus A. We do the same process. We We notice that there's a technically a plus here and a minus here. So if we take up the negative, actually get minus B plus A. And the cool thing about actors is that the community of property holds, which means you can move these to the other side. So it gives you hey minus baby and same thing like last time where quantity we have. What's this? It's really long, Better it goes down. We sense this negative. It just goes the opposite way. We said see wass or the distance of of the this wass twenty to my dream about twenty two. And so now the angle here is the same age bows before Angel here was the same as before. This is about nineteen. And so later on in exercise one point three one, there's a different approach. And so you don't have to use all this trigger trigonometry. You just have to know the components of each of these vectors, which would be a lot nicer.