WEBVTT
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DAVID JORDAN: Hello, and
welcome back to recitation.
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I'd like to work this problem
with you, in which we're
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going to use
determinants to compute
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the area of a parallelogram
sitting in a plane.
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So why don't you take
a moment to-- why
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don't you take some time to work
this out, and we'll check back
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and you can see how I did it.
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OK, so let's get
started on this problem.
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Now the first thing
that we need to be
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careful about with
this problem is,
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we know that we want
to take a determinant,
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but we need to be careful.
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Determinants of pairs
of vectors make sense.
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Determinants of points
do not make sense.
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So here we have
these four points,
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which are the endpoints
of the parallelogram.
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And what we need to do
from these four points
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is get some vectors that
we can compute with.
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So over here, I have
taken the vectors
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which connect the endpoints
of the parallelogram.
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So you'll see that this
[6, 1] here, this vector [6,
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1] is coming from the point
(1, 1) in the original
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parallelogram and (7, 2).
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So this vector [6, 1] is
just the difference of (7, 2)
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and the point (1, 1).
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And similarly [5, 2]
here is the difference
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of our original point (6, 3)
and our base point (1, 1).
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So now that we have
these two vectors,
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the area of our
parallelogram is just
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going to be the determinant
of our two vectors.
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Well, we'd better be careful.
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It's going to be plus or
minus the determinant,
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is going to be the area.
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So let's compute
this determinant.
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So we find 6 times 2 minus
5-- so we get 12 minus 5 is 7.
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Now we got a positive number,
and so this plus or minus
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we take to be positive.
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Had we computed our determinant
by transposing the rows here,
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then we might have
found a negative 7,
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and of course we want
our area to be positive,
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so we would just choose 7.
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So, let me just go through the
one tricky part of this problem
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is the original endpoints of
our parallelogram are not what
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are important for the area.
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What's important is
the vectors which
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connect the two of our
endpoints together.
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And so we computed those,
[6, 1] and [5, 2], and then
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taking their
determinant gives us
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the area of the parallelogram.
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OK, I'll leave it at that.