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University of North Texas

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Problem 22 Medium Difficulty

Find, correct to the nearest degree, the three angles of the triangle with the given vertices.

$ A (1, 0, -1) $ , $ B (3, -2, 0) $ , $ C (1, 3, 3) $

Answer

angle at vertex $\quad A : 97.7^{\circ} \quad B : 53.5^{\circ} \quad C : 28.8^{\circ}$

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Video Transcript

if we want to find the angles for this triangle here, Uh, the first thing we're going to need to do is go ahead and write out some factors because excuse me, we know if so, like, let's first just call this angle Here. Data this vector here going from a to B just gonna be a B and then the one going from a C s, A C. We know that if we take the dot product of this, this should be equal to the magnitudes of, uh, eight times the magnitude of a C times the co sign of the angle between them. And then if we want to solve for co sign or for not cause I'm but data we would divide over by the magnitudes and take coastline inverse. So that's going to give us data is equal to the dot product of a B with a c over each of their magnitudes. And then we just take cosine inverse of it. So let's first go ahead and figure out what is a B and a C. And then we'll at least have our angle Fada there. So to get a B, this is just going to be the point B minus the point because it's the and minus the start. So B is three negative. 20 a is 10 negative one, and then we're just going to track these component wise. So three minus one is two negative. Two minus zero is negative. Two zero minus negative one or the negatives cancel. So that would just be positive one now? Yeah. All right. So that's our vector from a to B to get a to see we're going to do C minus a. So c is 133 minus, um, a which is 10 negative one. So we have one minus one. That's 03 minus 033 minus minus one. That would be four. Because we're adding now we need to get the magnitudes of these. So let's just go ahead and take the magnitude of both of them right here. So remember to find the magnitude we square each of the components, Adama, then take the square root. So this is going to be two squared, plus negative two squared, plus one squared, all square rooted one. So it'd be four plus 48 uh, +19 and then Route nine is just three. The doubt here. So this would be zero squared plus three squared plus 16 squared. So that would be nine. Plus, I wrote 16 as opposed to four. I wondered why the number was so big. I was getting ahead of myself, so it would be nine plus 16, which is 25. And then Route 25 is five. So now all we need is the dot product between these, so a b dotted with a C is going to be well, we multiply these two components together. Or I should say, we multiply everything component wise together, and then we add the result, which so that would be negative. Two times three plus one times four. Okay. And so then this is zero. This is negative six. And this is, uh, four. So overall, they give us negative, too. So let me pick this up, then scooted down to here, and now we can just go ahead and plug in these numbers so we end up with our theater is going to be co sign inverse of the DOT product was negative, too. And then we have three times five down here. So that would be co sign in for some negative to 15th as data. But they wanted us to round this to one decimal place. So let's go ahead and do that. So negative two divided by 15 coastline immersed. Remember, when you're plugging this into your calculator, um, to have where this is in degrees, um or, I mean, you could have an ingredients, but just kind of make note of whichever one you're using. So we have data, and I'm just gonna round this up because it's 97.6. I'll write it up to 98 degrees. Yeah. Uh, now we can go ahead and repeat this process, but for another angle. So I'm going to find Let me screw this down some this angle here next. So let's call this alpha. So this is going to be if we follow the same kind of steps that we have over here. Alpha is eager to co sign inverse of well, here. We're going to have this factor going from B to C and then B to a So this is going to be B to C started with B to a all over each of their magnitudes. And one thing that's kind of nice is notice that the only difference between B A and A B is the direction in which they're facing. So be a is the same thing as negative a B. So if we come up here and find what we got for a B, I'm just going to be lazy and pick this up and scoot it down. We can just throw a negative out front here and then that gives us be a so b a is going to be negative. 22 negative one. And actually, a nice thing about it is that the magnitude of the veins are also going to be equal to each other. So what was the magnitude we got up here for? This was three. So I can just go ahead and write down three and we don't even have to worry about actually make sure that was a B. Yeah. Now we just need to go ahead and find for BC so B to C is going to be C minus. B, uh, see is 133 b is three negative 20 and again, we just go ahead and subtract component wise. So one minus three is negative. Two, uh, three minus minus two That's going to be five and then three minus zero is just three, and we can go ahead. Take the magnitude of this. So negative two squared plus five squared, plus three squared, all square rooted. So that's going to be four plus 25 plus nine, which is 38. So that would just be Route 38. And the last thing we need now is the dot product between each of these, uh, so be a dotted with the sea is well, we're going to do negative two times. Negative two So negative two times negative two. Then two times five plus two times five and then negative one times three plus negative one times three and then we just go ahead and add all of those up. So let's see, That's four. That's 10 negative three. So that should be 11. So let's come up here and scooch this down and then we can plug everything in. Well, actually, I plug it in right here. So the magnitude of this is three. That's Route 38 to 3, Route 38 all over and then we had our dot product of 11, right? And actually, I might as well just shoot that down. So I have more room to write car. And if we plug this into a calculator now, So 11 divided by 3 10 11, divided by three, divided by route 38 and co sign and diverse. So this is saying that alpha is approximately so it would be 53.5, but we're rounding it to the nearest degrees Would be 54. So let me go ahead and scoot my drawing down one more time. And now we found that this angle here Alpha is 54. And actually, what did we get for theta earlier? 98. So let me just go ahead and write this down here. Now. We could go ahead if we wanted. And mhm. Excuse me, um, to find our last angle. What we could do is so let's just call this gamma, and we could go through the same procedure we just did, but that's going to take a bit of time. So what you might recall is that all of the angles of a triangle should add up to 180 degrees. So we just plug in. Eight is 98. Alpha is 54 then solve for gamma. So gamma is just going to be No. 1, 80 minus 54 minus 98. And so subtracting that would give us 28. So I'll just write that over here. So gamma is approximately 28 degrees, so these would be our three angles, and that should be approximately right there.