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