In Exercises 17-22, find the angle between and W. Round to the nearest tenth of & degree 17. v = 2i - j w = 3i + 4j 18. = -2i + 5j. w = 3i + 6j 19. = ~3i + 2j; ...

In Exercises 17–22, find the angle between $v$ and $w$ . Round to the nearest tenth of a degree.
$$\mathbf{v}=-2 \mathbf{i}+5 \mathbf{j}, \quad \mathbf{w}=3 \mathbf{i}+6 \mathbf{j}$$

All right. So we need to find the angle between two factors for V and W R General form is gonna be the angle equal to the inverse co sign of this right here. So basically what we're gonna do again fines this We're gonna find out. Be times w divided by the magnitude of the times. W the inverse code. Sign of the heart. First, let's find the magnitude of the vehicle square roots, Megan's three Word plus to sway square root two, not the magnitude W Square Roots four squared was negative. One. Where is this gonna be? Square root of seven. Next. Let's find the dot product between B and W. So remember our general forms a one times a two plus B one times be too. So we have negative three times four waas, two times negative one which is gonna be equal to negative. This is dot product to be now in the top. In the numerator, we have negative 14 and then in the bottom, we have square roots, 13 times square roots 17. So we're gonna find this number right here, and then take the inverse coaster sign of its on our couch. So, after plugging this enjoyed calculator, get the angle between these two equal to 160 points three

Okay, So for this problem, when we're gonna find the angle between V and W and round the nearest 10th of a degree so we're gonna be looking for theta, which is the angle, and we're gonna use the equation. Coastline data equals V times w divided by the magnitude of V times the magnitude of w. So some important things to know is when we see the times w at the top right there any times w we we know that that's the dot product. Um, so I'm just gonna put the piece so we can note that. And so the things we have to solve for before plugging into this equation as we need to find the dot product of the times w the magnitude of e and the magnitude of w So let's start with the dot product at the top. So the dot product V terms W is gonna be equal to the one that'll be you one plus V two w two. So let's figure out what the one w wanted. We're gonna be etcetera. So we're gonna look at what they give us for what V and w equal. So the first number with the first variable, which is eye on V is one. So any time we see variable by itself, we just assume that there's a one in front of it is one times the variable is the variable. So one is gonna be V one, and our next number's too with associated with the second variable J so too is gonna be V two. And then we're looking at W with same idea. So four is associated with the eye. So that's gonna be W one. And negative three is associated with the J. So negative three is gonna be W two. So now we just have to play that in. So the terms w is equal to be one which we said was one multiplied by W one, which we said was four and then plus V two, which is to multiplied by W two, which is negative. Three. So now we just have to simplify that we get, um, four right here, plus negative six, which is minus six. Well, I'm just gonna write this down here so we don't get lost in our work. So the times w equals four minus six, which simplifies down to negative too So now we have that first part of that equation that we need. Now we need to find the magnitude of E and the magnitude of W. So let's start with the first. So the magnitude of the I was going to equal the square root V one squared, plus the two squared. So it's plugging those values So magnitude of e is equal Thio square root of the one. I'm sorry we redo that. It's gonna be equal to the square root of the one which is one so one squared plus V two squared V two is +22 squared and if we simplify it, that's gonna be equal. Thio, the square root of one plus four She was gonna equal skirt of five. And now we need to find the magnitude of W We're gonna take a similar approach. The the magnitude of W is going to equal square root. Uh, w one squared plus W two squared. All right, now, let's put this in. So magnitude of W is equal to the square root of W one squared, which is four. So four squared. Plus, don't be too squared, which is going to be negative. Three really important to include parentheses here, since we have a negative number. So if we simplify this, we're going to get the magnitude of W Equal Skirt of four Squared, which is 16 plus the plus negative three squared, which isn't positive. Nine. If we do 16 plus nine, we're gonna get the screw it of 25 on describable 25. It's a simple square, so that's just gonna be five. So now let's play this in. So you co signed data and we already figured out the dot product of V times W, which is negative, too. So we're gonna plug that in right there, divided by the magnitude of V, which we said was far of squared. If I excuse me, multiplied by the magnitude of W, which we said was five. And now we just need to solve fourth data. So to sell for theater, we're gonna do in verse co sign on both sides. It's a little messy, but I'm gonna rewrite that step. So that's gonna leave us with The ADA is equal to universe co sign of negative too, right? Goodbye. Five square root. It's all right. So you're gonna plug that into your calculator and you're gonna get beta two equal. 100 0.3 agrees

So we need to find the angle between the two factors V and w r General equation for this is gonna be the inverse coast on the dot product of BMW divided by the magnitude of the times magnitude of w Basically what we're gonna dio is gonna find this number right here and then plug it in our calculators for the inverse coast. So first, let's find the magnitude of each so the magnitude of V gonna be equal Thio square root of you squared waas negative one and this is gonna be equal to the square root of five magnitude w gonna be equal to square roots really squared plus four squared which is gonna be equal to five Now that we have both magnitudes, we need to find the dot product of BMW dot w gonna be equal Thio two times three. Remember, we're using your general formula a one times a two plus B one times plus negative four are positive for times and negative one. This is gonna be equal to two. Now we have two divided by five times square root of five. I'm not going to take the universe coast side of this this is gonna be equal to the angle between the two factors. So you take this number and then you're gonna plug it into calculated for the inverse co sign. You're gonna get data. The angle between the two factors is equal to 79 point six nine five.

Okay, so this problem is to find the angle to envy and w and round to the nearest 10th of a degree. All right, so let's point out what we're looking for. We are looking for the angle, which is the same as beta. So that's what we're looking for here. Um, we're looking at this equation. It's important to note that many times, don't you? Right there. We know that that's the dot product, and I'm gonna put DP just so we know that's the dot product that we're gonna be working with. Okay, so first things first, let's also Okay, well, the equation is co sign theta equals V times W So the doc product of you Time Sylvia fighter by the magnitude of E and the magnitude of duck you. So before plugging into the equation, we're gonna find what v times w equals with the magnitude of e equals and the magnitude of W. So let's start with the dot product of you times W. Okay, so the dot product of the terms W we know is gonna be equal to be one w one post view too. W two so Well, it's It's for your V one, w one v two w tour. So we look at our equations we have or what? What V and w the vectors were given equal. We have equals three J and w equals four. I post fact j So to figure out what the one of you two are, let's call Liar, First Variable and J R. Second very well important thing to know is that I is not included, and v So we need to write. Rewrite that. So any time you have zero multiplied brain number, we know that equal zero So V equals three J's the same a saying V equals zero I plus three j Okay, so if we do that, we know that right there zero is going to be V one and the three right there is going to be the two. Sorry for that. Getting a little messy. And if we look at W R first number with the first variables for so four is gonna be w one and four, I was gonna be w two. We get rid of that blue space. Okay, so now that we have that, we can plug it in. So the times W is going to equal zero multiplied by W one, which is for plus the two, which is three multiplied by W two or just five. So if we simplify this, the zero on the four gonna cancel out and were left with three times five, which is equal to 15. So now we have that first dot product that we need to find. Now we need to figure out what the magnitude of e and the magnitude of W. R. So let's start with the magnitude of the I'm gonna put it in a different color. So the man into T. V is gonna be equal to the square root of the one squared plus ze to squared. Okay, so we already figured out. What do you want to be too? Are So now we can just plug it in so magnitude of e is gonna be equal to the square root of V one, which we said with zero squared plus V two squared, so would be to his three. So three squared we know, zero squared to zero. That's light cancels out and we're left with the magnitude of the equal to the square root of three squared. Sorry, that's a two. So the square root of three squared is gonna equal square root of nine, which is equal to three. All right now we need to figure out the magnitude of W, so let's put that right below V. So the magnitude of W it's going to equal square root of w one squared plus W two squared. So let's put that in. So magnitude of W was equal to W one, which is r squared of W one squared, which is four squared plussed of you two squared, just five squared. And if we simplify that, that gets us square root. So 16 plus 25 well, under the step radical right there, which is equal to the square root of four. You won. Okay, so now we have the magnitude of the the magnitude of W and the dot product of the times W So now we can plug him, so co sign, they'd, uh, it's gonna be equal. Thio be times W, which they said was 15 decided by the magnitude of the which we said was three right there multiplied by the magnitude of W, which you said was square to 41 and then we need to solve for theta again. So inverse co sign on both sides. Okay? And we're gonna be left with Sita equals inverse co sign of whoops. Sorry about that. Off big parentheses. 15 decided by three square Route 41. So now we can plug this into our calculators and we're gonna get data equal Thio 38.7 degrees.