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University of North Texas

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Problem 55 Medium Difficulty

Find the angle between a diagonal of a cube and one of its edges.

Answer

$54.74^{\circ}$

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Video Transcript

So if we want to find the angle between the diagonal and one of its edges, let's just go ahead and draw that. And so I'll use this diagonal here, so I shall just draw it. Ah, and then we have this here. So if we want to find whatever this angle is, what we need to do is come up with some kind of coordinate system so we can figure out these vectors. Um, So what I'm gonna do is I'm just going to call this the origin here and then all say, going out this way. Is this going to be one, or actually, we'll just say a because I'll say all of these have a side length of a So at this point here is just gonna be a 00 So the vector going from the origin to this is just going to be AIDS or zero. So you already have that. And then the one up top here, well, we go a over a up and a out like that. So this is going to be a and you could just use a number, but you'll see that we'll get the same regardless. So what we can go ahead and do to start is first. Take the dot product of these because I'll call this X and I'll call this Why. So let me expand the screen because we know X started with fly is going to be the magnitude of x times, the magnitude of y times, the coastline of the angle between them And then we can divide the magnitudes over. Take art co sign on each side So we get coastline Inverse of ex daughter will fly all over the magnitude of x times the magnitude of why that would be our angle. So let's go ahead and just find these magnitudes first. So the magnitude here remember, we just square everything. Add it all up so that would be a square, plus a squared plus a squared ah, square rooted. So that would just be the square root of three a square which would just be a Route three. And then over here. Uh, if we take the magnitude of this well, every component is just going to be zero outside of a. So that's really just finding the length of a So that's a little easy. There's this day machine Now what we're going to need to do is find the dot product between these. So let me go ahead and write that down here. So we have a dotted with a 00 And remember, we multiply them component wise and then add the results. So this is going to be equal to it would be a squared. Plus, we'll eight times zero plus eight times zero again is zero. So that's just going to be a squared so we can come back up here now. I didn't mean to do that and just plug everything in. So the co sign inverse of a squared over a over route three. Well, actually, it be a times a times Route three almost forgot that so that a squares can sell. And this is why I was saying it doesn't really matter if we use it or not, but, um, never hurts to be more general. Just one over root three. So this here is the exact solution. Um, and if we want an approximate because I honestly have no idea what that is supposed to be, we can go ahead and just plug this into our calculators, and we'll get something around 54 point seven degrees. So either these would be valid solutions