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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 going

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to use determinants to compute
the area of a parallelogram

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sitting in a plane.

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So why don't you take a moment
to, why don't you take some

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time to work this out, and we'll
check back and you can

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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 careful about with

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this problem is, 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, which are the

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endpoints of the
parallelogram.

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And what we need to do from
these four points is get some

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vectors that we can
compute with.

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So over here, I have taken the
vectors which connect the

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endpoints of the
parallelogram.

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So you'll see that this 6, 1
here, this vector 6, 1 is

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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 and the

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point 1, 1.

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And similarly 5, 2 here is the
difference of our original

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point 6, 3 and our
base point 1, 1.

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So now that we have these two
vectors, the area of our

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parallelogram is just
going to be the

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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 is

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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--

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so we get 12 minus 5--

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is 7.

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Now we got a positive number,
and so this plus or minus we

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take to be positive.

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Had we computed our determinant
by transposing the

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rows here, then we might have
found a negative seven, and of

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course we want our area to be
positive, so we would just

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choose seven.

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So, let me just go through the
one tricky part of this

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problem is the original
endpoints of our parallelogram

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are not what are important
for the area.

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What's important is the vectors
which connect the two

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of our endpoints together.

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And so we computed those 6, 1
and 5, 2, and then taking

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their determinant gives us the
area of the parallelogram.

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OK, I'll leave it at that.

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