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Let me just tell you first
about the list of topics.

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Basically, the list of topics
is simple.

00:00:35.000 --> 00:00:42.000
It is everything.
I mean, everything we have seen

00:00:42.000 --> 00:00:48.000
so far is on the exam.
But let me just remind you of

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the main topics that we have
seen.

00:00:52.000 --> 00:00:58.000
First of all,
we learned about vectors,

00:00:58.000 --> 00:01:03.000
how to use them,
and dot-product.

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At this point,
you probably should know that

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the dot-product of two vectors
is obtained by summing products

00:01:13.000 --> 00:01:19.000
of components.
And geometrically it is the

00:01:19.000 --> 00:01:29.000
length of A times the length of
B times the cosine of the angle

00:01:29.000 --> 00:01:34.000
between them.
And, in particular,

00:01:34.000 --> 00:01:41.000
we can use dot-product to
measure angles by solving for

00:01:41.000 --> 00:01:49.000
cosine theta in this equality.
And most importantly to detect

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whether two vectors are
perpendicular to each other.

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Two vectors are perpendicular
when their dot-product is zero.

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Any questions about that?
No.

00:02:07.000 --> 00:02:15.000
Is everyone reasonably happy
with dot-product by now?

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I see a stunned silence.
Nobody happy with dot-product

00:02:19.000 --> 00:02:28.000
so far?
OK.

00:02:28.000 --> 00:02:36.000
If you want to look at Practice
1A, a good example of a typical

00:02:36.000 --> 00:02:41.000
problem with dot-product would
be problem 1.

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Let's see.
We are going to go over the

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practice exam when I am done
writing this list of topics.

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I think probably we actually
will skip this problem because I

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think most of you know how to do
it.

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And if not then you should run
for help from me or from your

00:02:57.000 --> 00:03:02.000
recitation instructor to figure
out how to do it.

00:03:02.000 --> 00:03:10.000
The second topic that we saw
was cross-product.

00:03:10.000 --> 00:03:19.000
When you have two vectors in
space, you can just form that

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cross-product by computing its
determinant.

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So, implicitly,
we should also know about

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determinants.
By that I mean two by two and

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three by three.
Don't worry about larger ones,

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even if you are interested,
they won't be on the test.

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And applications of
cross-product,

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for example,
finding the area of a triangle

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or a parallelogram in space.
If you have a triangle in space

00:04:04.000 --> 00:04:10.000
with sides A and B then its area
is one-half of the length of A

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cross B.
Because the length of A cross B

00:04:14.000 --> 00:04:19.000
is length A, length B sine
theta, which is the same as the

00:04:19.000 --> 00:04:24.000
area of the parallelogram formed
by these two vectors.

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And the other application of
cross-product is to find a

00:04:28.000 --> 00:04:32.000
vector that's perpendicular to
two given vectors.

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In particular,
to find the vector that is

00:04:36.000 --> 00:04:42.000
normal to a plane and then find
the equation of a plane.

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Another application is finding
the normal vector to a plane and

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using that finding the equation
of a plane.

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Basically, remember,
to find the equation of a

00:05:16.000 --> 00:05:23.000
plane, ax by cz = d,
what you need is the normal

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vector to the plane.
And the components of the

00:05:29.000 --> 00:05:32.000
normal vector are exactly the
coefficients that go into this.

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And we have seen an argument
for why that happens to be the

00:05:38.000 --> 00:05:40.000
case.
To find a normal vector to a

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plane typically what we will do
is take two vectors that lie in

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the plane and will take their
cross-product.

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And the cross-product will
automatically be perpendicular

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to both of them.
We are going to see an example

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of that when we look at problem
5 in practice 1A.

00:06:04.000 --> 00:06:12.000
I think we will try to do that
one.

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Another application,
well, we will just mention it

00:06:17.000 --> 00:06:21.000
as a topic that goes along with
this one.

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We have seen also about
equations of lines and how to

00:06:34.000 --> 00:06:43.000
find where a line intersects a
plane.

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Just to refresh your memories,
the equation of a line,

00:06:46.000 --> 00:06:50.000
well, we will be looking at
parametric equations.

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To know the parametric equation
of a line, we need to know a

00:06:54.000 --> 00:06:59.000
point on the line and we need to
know a vector that is parallel

00:06:59.000 --> 00:07:03.000
to the line.
And, if we know a point on the

00:07:03.000 --> 00:07:06.000
line and a vector along the
line,

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then we can express the
parametric equations for the

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motion of a point that is moving
on the line.

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Actually, starting at point,
at time zero,

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and moving with velocity v.

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To put things in symbolic form,
you will get a position of that

00:07:44.000 --> 00:07:48.000
point by starting with a
position of time zero and adding

00:07:48.000 --> 00:07:53.000
t times the vector v.
It gives you x,

00:07:53.000 --> 00:08:00.000
y and z in terms of t.
And that is how we represent

00:08:00.000 --> 00:08:06.000
lines.
We will look at problem 5 in a

00:08:06.000 --> 00:08:15.000
bit, but any general questions
about these topics?

00:08:15.000 --> 00:08:20.000
No.
Do you have a question?

00:08:20.000 --> 00:08:26.000
Do we have to know Taylor
series?

00:08:26.000 --> 00:08:34.000
That is a good question.
No, not on the exam.

00:08:34.000 --> 00:08:36.000
[APPLAUSE]
Taylor series are something you

00:08:36.000 --> 00:08:38.000
should be aware of,
generally speaking.

00:08:38.000 --> 00:08:40.000
It will be useful for you in
real life,

00:08:40.000 --> 00:08:44.000
probably not when you go to the
supermarket,

00:08:44.000 --> 00:08:48.000
but if you solve engineering
problems you will need Taylor

00:08:48.000 --> 00:08:52.000
series.
It would be good not to forget

00:08:52.000 --> 00:08:58.000
them entirely,
but on the 18.02 exams they

00:08:58.000 --> 00:09:03.000
probably won't be there.
Let me continue with more

00:09:03.000 --> 00:09:06.000
topics.
And then we can see if you can

00:09:06.000 --> 00:09:11.000
think of other topics that
should or should not be on the

00:09:11.000 --> 00:09:22.000
exam.
Third topics would be matrices,

00:09:22.000 --> 00:09:34.000
linear systems,
inverting matrices.

00:09:34.000 --> 00:09:37.000
I know that most of you that
have calculators that can invert

00:09:37.000 --> 00:09:40.000
matrices, but still you are
expected at this point to know

00:09:40.000 --> 00:09:42.000
how to do it by hand.
If you have looked at the

00:09:42.000 --> 00:09:44.000
practice tests,
both of them have a problem

00:09:44.000 --> 00:09:47.000
that asks you to invert a matrix
or at least do part of it.

00:09:47.000 --> 00:09:53.000
And so it is very likely that
tomorrow there will be a problem

00:09:53.000 --> 00:09:57.000
like that as well.
In general, when a kind of

00:09:57.000 --> 00:10:00.000
problem is on both practice
tests it's a good indication

00:10:00.000 --> 00:10:04.000
that it might be there also on
the actual exam.

00:10:04.000 --> 00:10:08.000
Unfortunately,
not with the same matrix so you

00:10:08.000 --> 00:10:12.000
cannot learn the answer by
heart.

00:10:12.000 --> 00:10:17.000
Another thing that we have
learned about,

00:10:17.000 --> 00:10:24.000
well, I should say this is
going to be problem 3 on the

00:10:24.000 --> 00:10:29.000
test and will on the practice
test.

00:10:29.000 --> 00:10:33.000
On the actual test,
too, I think,

00:10:33.000 --> 00:10:37.000
actually.
Anyway, we will come back to it

00:10:37.000 --> 00:10:39.000
later.
A couple of things that you

00:10:39.000 --> 00:10:42.000
should remember.
If you have a system of the

00:10:42.000 --> 00:10:46.000
form AX equals B then there are
two cases.

00:10:46.000 --> 00:10:51.000
If a determinant of A is not
zero then that means you can

00:10:51.000 --> 00:10:56.000
compute the inverse matrix and
you can just solve by taking A

00:10:56.000 --> 00:11:02.000
inverse times B.
And the other case is when the

00:11:02.000 --> 00:11:11.000
determinant of A is zero,
and there is either no solution

00:11:11.000 --> 00:11:18.000
or there is infinitely many
solutions.

00:11:18.000 --> 00:11:20.000
In particular,
if you know that there is a

00:11:20.000 --> 00:11:21.000
solution,
for example,

00:11:21.000 --> 00:11:24.000
if B is zero there always is an
obvious solution,

00:11:24.000 --> 00:11:28.000
X equals zero,
then you will actually have

00:11:28.000 --> 00:11:30.000
infinitely many.
In general, we don't really

00:11:30.000 --> 00:11:33.000
know how to tell whether it is
no solution or infinitely many.

00:11:43.000 --> 00:11:52.000
Questions about that?
Yes?

00:11:52.000 --> 00:11:58.000
Will we have to know how to
rotate vectors and so on?

00:11:58.000 --> 00:12:01.000
Not in general,
but you might still want to

00:12:01.000 --> 00:12:04.000
remember how to rotate a vector
in a plane by 90 degrees because

00:12:04.000 --> 00:12:07.000
that has been useful when we
have done problems about

00:12:07.000 --> 00:12:10.000
parametric equations,
which is what I am coming to

00:12:10.000 --> 00:12:12.000
next.
What we have seen about

00:12:12.000 --> 00:12:15.000
rotation matrices,
that was the homework part B

00:12:15.000 --> 00:12:17.000
problem,
you are not supposed to

00:12:17.000 --> 00:12:21.000
remember by heart everything
that was in part B of your

00:12:21.000 --> 00:12:24.000
homework.
It is a good idea to have some

00:12:24.000 --> 00:12:27.000
vague knowledge because it is
useful culture,

00:12:27.000 --> 00:12:28.000
I would say,
useful background for later in

00:12:28.000 --> 00:12:33.000
your lives,
but I won't ask you by heart to

00:12:33.000 --> 00:12:40.000
know what is the formation for a
rotation matrix.

00:12:40.000 --> 00:12:49.000
And then we come to,
last by not least,

00:12:49.000 --> 00:13:00.000
the problem of finding
parametric equations.

00:13:00.000 --> 00:13:03.000
And, in particular,
possibly by decomposing the

00:13:03.000 --> 00:13:06.000
position vector into a sum of
simpler vectors.

00:13:06.000 --> 00:13:13.000
You have seen quite an evil
exam of that on the last problem

00:13:13.000 --> 00:13:19.000
set with this picture that maybe
by now you have had some

00:13:19.000 --> 00:13:23.000
nightmares about.
Anyway, the one on the exam

00:13:23.000 --> 00:13:27.000
will certainly be easier than
that.

00:13:27.000 --> 00:13:36.000
But, as you have seen -- I
mean, you should know,

00:13:36.000 --> 00:13:40.000
basically,
how to analyze a motion that is

00:13:40.000 --> 00:13:44.000
being described to you and
express it in terms of vectors

00:13:44.000 --> 00:13:48.000
and then figure out what the
parametric equation will be.

00:13:48.000 --> 00:13:52.000
Now, again, it won't be as
complicated on the exam as the

00:13:52.000 --> 00:13:57.000
one in the problem set.
But there are a couple of those

00:13:57.000 --> 00:14:01.000
on the practice exam,
so that gives you an idea of

00:14:01.000 --> 00:14:04.000
what is realistically expected
of you.

00:14:04.000 --> 00:14:09.000
And now once we have parametric
equations for motion,

00:14:09.000 --> 00:14:13.000
so that means when we know how
to find the position vector as a

00:14:13.000 --> 00:14:15.000
function of a parameter maybe of
time,

00:14:15.000 --> 00:14:24.000
then we have seen also about
velocity and acceleration,

00:14:24.000 --> 00:14:28.000
which the vector is obtained by
taking the first and second

00:14:28.000 --> 00:14:31.000
derivatives of a position
vector.

00:14:31.000 --> 00:14:41.000
And so one topic that I will
add in there as well is somehow

00:14:41.000 --> 00:14:50.000
how to prove things about
motions by differentiating

00:14:50.000 --> 00:14:55.000
vector identities.
One example of that,

00:14:55.000 --> 00:14:58.000
for example,
is when we try to look at

00:14:58.000 --> 00:15:03.000
Kepler's law in class last time.
We look at Kepler's second law

00:15:03.000 --> 00:15:06.000
of planetary motion,
and we reduced it to a

00:15:06.000 --> 00:15:10.000
calculation about a derivative
of the cross-product R cross v.

00:15:10.000 --> 00:15:14.000
Now, on the exam you don't need
to know the details of Kepler's

00:15:14.000 --> 00:15:16.000
law,
but you need to be able to

00:15:16.000 --> 00:15:22.000
manipulate vector quantities a
bit in the way that we did.

00:15:22.000 --> 00:15:27.000
And so on practice exam 1A,
you actually have a variety of

00:15:27.000 --> 00:15:30.000
problems on this topics because
you have problems two,

00:15:30.000 --> 00:15:36.000
four and six,
all about parametric motions.

00:15:36.000 --> 00:15:42.000
Probably tomorrow there will
not be three distinct problems

00:15:42.000 --> 00:15:48.000
about parametric motions,
but maybe a couple of them.

00:15:48.000 --> 00:15:51.000
I think that is basically the
list of topics.

00:15:51.000 --> 00:15:54.000
Anybody spot something that I
have forgotten to put on the

00:15:54.000 --> 00:15:58.000
exam or questions about
something that should or should

00:15:58.000 --> 00:16:03.000
not be there?
You go first.

00:16:03.000 --> 00:16:11.000
Yeah?
How about parametrizing weird

00:16:11.000 --> 00:16:18.000
trigonometric functions?
I am not sure what you mean by

00:16:18.000 --> 00:16:21.000
that.
Well, parametric curves,

00:16:21.000 --> 00:16:25.000
you need to know how to
parameterize motions,

00:16:25.000 --> 00:16:30.000
and that involves a little bit
of trigonometrics.

00:16:30.000 --> 00:16:33.000
When we have seen these
problems about rotating wheels,

00:16:33.000 --> 00:16:34.000
say the cycloid,
for example,

00:16:34.000 --> 00:16:38.000
and so on there is a bit of
cosine and sine and so on.

00:16:38.000 --> 00:16:41.000
I think not much more on that.
You won't need obscure

00:16:41.000 --> 00:16:48.000
trigonometric identities.
You're next.

00:16:48.000 --> 00:16:51.000
Any proofs on the exam or just
like problems?

00:16:51.000 --> 00:16:55.000
Well, a problem can ask you to
show things.

00:16:55.000 --> 00:16:58.000
It is not going to be a
complicated proof.

00:16:58.000 --> 00:17:00.000
The proofs are going to be
fairly easy.

00:17:00.000 --> 00:17:04.000
If you look at practice 1A,
the last problem does have a

00:17:04.000 --> 00:17:08.000
little bit of proof.
6B says that show that blah,

00:17:08.000 --> 00:17:11.000
blah, blah.
But, as you will see,

00:17:11.000 --> 00:17:13.000
it is not a difficult kind of
proof.

00:17:13.000 --> 00:17:24.000
So, about the same.
Yes?

00:17:24.000 --> 00:17:29.000
Are there equations of 3D
shapes that we should know at

00:17:29.000 --> 00:17:32.000
this point?
We should know definitely a lot

00:17:32.000 --> 00:17:34.000
about the equations of planes on
lines.

00:17:34.000 --> 00:17:37.000
And you should probably know
that a sphere centered at the

00:17:37.000 --> 00:17:40.000
origin is the set of points
where distance to the center is

00:17:40.000 --> 00:17:43.000
equal to the radius of the
sphere.

00:17:43.000 --> 00:17:45.000
We don't need more at this
point.

00:17:45.000 --> 00:17:48.000
As the semester goes on,
we will start seeing cones and

00:17:48.000 --> 00:17:52.000
things like that.
But at this point planes, lines.

00:17:52.000 --> 00:17:58.000
And maybe you need to know
about circles and spheres,

00:17:58.000 --> 00:18:03.000
but nothing beyond that.
More questions?

00:18:03.000 --> 00:18:11.000
Yes?
If there is a formula that you

00:18:11.000 --> 00:18:14.000
have proved on the homework
then, yes, you can assume it on

00:18:14.000 --> 00:18:17.000
the test.
Maybe you want to write on your

00:18:17.000 --> 00:18:20.000
test that this is a formula you
have seen in homework just so

00:18:20.000 --> 00:18:24.000
that we know that you remember
it from homework and not from

00:18:24.000 --> 00:18:27.000
looking over your neighbor's
shoulder or whatever.

00:18:27.000 --> 00:18:31.000
Yes, it is OK to use things
that you know general-speaking.

00:18:31.000 --> 00:18:33.000
That being said,
for example,

00:18:33.000 --> 00:18:37.000
probably there will be a linear
system to solve.

00:18:37.000 --> 00:18:40.000
It will say on the exam you are
supposed to solve that using

00:18:40.000 --> 00:18:43.000
matrices, not by elimination.
There are things like that.

00:18:43.000 --> 00:18:47.000
If a problem says solve by
using vector methods,

00:18:47.000 --> 00:18:51.000
things like that,
then try to use at least a

00:18:51.000 --> 00:18:54.000
vector somewhere.
But, in general,

00:18:54.000 --> 00:18:58.000
you are allowed to use things
that you know.

00:18:58.000 --> 00:19:07.000
Yes?
Will we need to go from

00:19:07.000 --> 00:19:09.000
parametric equations to xy
equations?

00:19:09.000 --> 00:19:16.000
Well, let's say only if it is
very easy.

00:19:16.000 --> 00:19:19.000
If I give you a parametric
curve, sin t,

00:19:19.000 --> 00:19:25.000
sin t, then you should be able
to observe that it is on the

00:19:25.000 --> 00:19:29.000
line y equals x,
not beyond that.

00:19:29.000 --> 00:19:37.000
Yes?
Do we have to use -- Yes.

00:19:37.000 --> 00:19:40.000
I don't know if you will have
to use it, but certainly you

00:19:40.000 --> 00:19:43.000
should know a little bit about
the unit tangent vector.

00:19:43.000 --> 00:19:51.000
Just remember the main thing to
know that the unit tangent

00:19:51.000 --> 00:19:57.000
vector is velocity divided by
the speed.

00:19:57.000 --> 00:20:04.000
I mean there is not much more
to it when you think about it.

00:20:04.000 --> 00:20:11.000
Yes?
Kepler's law,

00:20:11.000 --> 00:20:13.000
well, you are allowed to use it
if it helps you,

00:20:13.000 --> 00:20:16.000
if you find a way to squeeze it
in.

00:20:16.000 --> 00:20:20.000
You don't have to know Kepler's
law in detail.

00:20:20.000 --> 00:20:22.000
You just have to know how to
reproduce the general steps.

00:20:22.000 --> 00:20:28.000
If I tell you R cross v is
constant, you might be expected

00:20:28.000 --> 00:20:33.000
to know what to do with that.
I would say -- Basically,

00:20:33.000 --> 00:20:36.000
you don't need to know Kepler's
law.

00:20:36.000 --> 00:20:38.000
You need to know the kind of
stuff that we saw when we

00:20:38.000 --> 00:20:41.000
derived it such as how to take
the derivative of a dot-product

00:20:41.000 --> 00:20:52.000
or a cross-product.
That is basically the answer.

00:20:52.000 --> 00:20:55.000
I don't see any questions
anymore.

00:20:55.000 --> 00:21:03.000
Oh, you are raising your hand.
Yes.

00:21:03.000 --> 00:21:06.000
How to calculate the distance
between two lines and the

00:21:06.000 --> 00:21:08.000
distance between two planes?
Well, you have seen,

00:21:08.000 --> 00:21:10.000
probably recently,
that it is quite painful to do

00:21:10.000 --> 00:21:13.000
in general.
And, no, I don't think that

00:21:13.000 --> 00:21:16.000
will be on the exam by itself.
You need to know how to compute

00:21:16.000 --> 00:21:19.000
the distance between two points.
That certainly you need to know.

00:21:19.000 --> 00:21:24.000
And also maybe how to find the
compliment of a vector in a

00:21:24.000 --> 00:21:28.000
certain direction.
And that is about it,

00:21:28.000 --> 00:21:33.000
I would say.
I mean the more you know about

00:21:33.000 --> 00:21:38.000
things the better.
Things that come up on part Bs

00:21:38.000 --> 00:21:43.000
of the problem sets are
interesting things,

00:21:43.000 --> 00:21:48.000
but they are usually not needed
on the exams.

00:21:48.000 --> 00:21:53.000
If you have more questions then
you are not raising your hand

00:21:53.000 --> 00:21:55.000
high enough for me to see it.
OK.

00:21:55.000 --> 00:22:00.000
Let's try to do a bit of this
practice exam 1A.

00:22:00.000 --> 00:22:03.000
Hopefully, everybody has it.
If you don't have it,

00:22:03.000 --> 00:22:07.000
hopefully your neighbor has it.
If you don't have it and your

00:22:07.000 --> 00:22:09.000
neighbor doesn't have it then
please raise your hand.

00:22:09.000 --> 00:22:10.000
I have a couple.

00:22:57.000 --> 00:23:02.000
If you neighbor has it then
just follow with them for now.

00:23:02.000 --> 00:23:05.000
I think there are a few people
behind you over there.

00:23:05.000 --> 00:23:06.000
I will stop handing them out
now.

00:23:06.000 --> 00:23:11.000
If you really need one,
it is on the website,

00:23:11.000 --> 00:23:15.000
it will be here at the end of
class.

00:23:15.000 --> 00:23:23.000
Let's see.
Well, I think we are going to

00:23:23.000 --> 00:23:27.000
just skip problems 1 and 2
because they are pretty

00:23:27.000 --> 00:23:32.000
straightforward and I hope that
you know how to do them.

00:23:32.000 --> 00:23:37.000
I mean I don't know.
Let's see.

00:23:37.000 --> 00:23:42.000
How many of you have no problem
with problem 1?

00:23:42.000 --> 00:23:46.000
How many of you have trouble
with problem 1?

00:23:46.000 --> 00:23:50.000
OK.
How many of you haven't raised

00:23:50.000 --> 00:23:52.000
your hands?
OK.

00:23:52.000 --> 00:23:56.000
How many of you have trouble
with problem 2?

00:23:56.000 --> 00:23:58.000
OK.
Well, if you have questions

00:23:58.000 --> 00:24:01.000
about those, maybe you should
just come see me at the end

00:24:01.000 --> 00:24:04.000
because that is probably more
efficient that way.

00:24:04.000 --> 00:24:11.000
I am going to start right away
with problem 3,

00:24:11.000 --> 00:24:18.000
actually.
Problem 3 says we have a matrix

00:24:18.000 --> 00:24:25.000
given to us |1 3 2;
2 0 - 1; 1 1 0|.

00:24:25.000 --> 00:24:30.000
And it tells us determinant of
A is 2 and inverse equals

00:24:30.000 --> 00:24:34.000
something, but we are missing
two values A and B and we are

00:24:34.000 --> 00:24:41.000
supposed to find them.
That means we need to do the

00:24:41.000 --> 00:24:51.000
steps of the algorithm to find
the inverse of A.

00:24:51.000 --> 00:25:00.000
We are told that A inverse is
one-half of |1 ...

00:25:00.000 --> 00:25:06.000
...; - 1 - 2 5;
2 2 - 6|.

00:25:06.000 --> 00:25:10.000
And here there are two unknown
values.

00:25:10.000 --> 00:25:15.000
Remember, to invert a matrix,
first we compute the minors.

00:25:15.000 --> 00:25:17.000
Then we flip some signs to get
the cofactors.

00:25:17.000 --> 00:25:19.000
Then we transpose.
And, finally,

00:25:19.000 --> 00:25:23.000
we divide by the determinant.
Let's try to be smart about

00:25:23.000 --> 00:25:26.000
this.
Do we need to compute all nine

00:25:26.000 --> 00:25:27.000
minors?
No.

00:25:27.000 --> 00:25:30.000
We only need to compute two of
them, right?

00:25:30.000 --> 00:25:34.000
Which minors do we need to
compute?

00:25:34.000 --> 00:25:39.000
Here and here or here and here?
Yeah, that looks better.

00:25:39.000 --> 00:25:43.000
Because, remember,
we need to transpose things so

00:25:43.000 --> 00:25:48.000
these two guys will end up here.
I claim we should compute these

00:25:48.000 --> 00:25:51.000
two minors.
And we will see if that is good

00:25:51.000 --> 00:25:52.000
enough.
If you start doing others and

00:25:52.000 --> 00:25:56.000
you find that they don't end up
in the right place then just do

00:25:56.000 --> 00:25:58.000
more,
but you don't need to spend

00:25:58.000 --> 00:26:00.000
your time computing all nine of
them.

00:26:00.000 --> 00:26:03.000
If you are worried about not
doing it right then,

00:26:03.000 --> 00:26:06.000
of course, you can maybe
compute one or two more to just

00:26:06.000 --> 00:26:11.000
double-check your answers.
But let us just do those that

00:26:11.000 --> 00:26:16.000
we think are needed.
The matrix of minors.

00:26:16.000 --> 00:26:22.000
The one that goes in the middle
position is obtained by deleting

00:26:22.000 --> 00:26:26.000
this row and that column,
and we are left with a

00:26:26.000 --> 00:26:32.000
determinant |3 2;1 0|,
3 times 0 minus 1 times 2

00:26:32.000 --> 00:26:40.000
should be - 2 should be - 2.
Then the one in the lower left

00:26:40.000 --> 00:26:45.000
corner, we delete the last row
and the first column,

00:26:45.000 --> 00:26:47.000
we are left with |3 2;
0 - 1|.

00:26:47.000 --> 00:26:50.000
3 times (- 1) is negative 3
minus 0.

00:26:50.000 --> 00:26:58.000
We are still left with negative
three.

00:26:58.000 --> 00:27:08.000
Is that step clear for everyone?
Then we need to go to cofactors.

00:27:08.000 --> 00:27:10.000
That means we need to change
signs.

00:27:10.000 --> 00:27:24.000
The rule is -- We change signs
in basically these four places.

00:27:24.000 --> 00:27:32.000
That means we will be left with
positive 2 and negative 3.

00:27:32.000 --> 00:27:44.000
Then we take the transpose.
That means the first column

00:27:44.000 --> 00:27:49.000
will copy into the first row,
so this guy we still don't

00:27:49.000 --> 00:27:52.000
know,
but here we will have two and

00:27:52.000 --> 00:27:59.000
here we will have minus three.
Finally, we have to divide by

00:27:59.000 --> 00:28:05.000
the determinant of A.
And here we are actually told

00:28:05.000 --> 00:28:08.000
that the determinant of A is
two.

00:28:08.000 --> 00:28:12.000
So we will divide by two.
But there is only one-half here

00:28:12.000 --> 00:28:18.000
so actually it is done for us.
The values that we will put up

00:28:18.000 --> 00:28:22.000
there are going to be 2 and
negative 3.

00:28:30.000 --> 00:28:38.000
Now let's see how we use that
to solve a linear system.

00:28:38.000 --> 00:28:46.000
If we have to solve a linear
system, Ax equals B,

00:28:46.000 --> 00:28:50.000
well, if the matrix is
invertible, its determinant is

00:28:50.000 --> 00:28:54.000
not zero,
so we can certainly write x

00:28:54.000 --> 00:29:01.000
equals A inverse B.
So we have to multiply,

00:29:01.000 --> 00:29:09.000
that is one-half | 1 2 - 3;
- 1 - 2 5; 2 2 - 6|.

00:29:09.000 --> 00:29:14.000
Times B [
1, - 2,1].

00:29:14.000 --> 00:29:17.000
Remember, to do a matrix
multiplication you take the rows

00:29:17.000 --> 00:29:21.000
in here, the columns in here and
you do dot-products.

00:29:21.000 --> 00:29:25.000
The first entry will be one
times one plus two times minus

00:29:25.000 --> 00:29:30.000
two plus minus three times one,
one minus four minus three

00:29:30.000 --> 00:29:35.000
should be negative six,
except I still have,

00:29:35.000 --> 00:29:43.000
of course, a one-half in front.
Then minus one plus four plus

00:29:43.000 --> 00:29:50.000
five should be 8.
Two minus four minus six should

00:29:50.000 --> 00:29:57.000
be -8.
That will simplify to [- 3,4,

00:29:57.000 --> 00:30:04.000
- 5].
Any questions about that?

00:30:04.000 --> 00:30:08.000
OK.
Now we come to part C which is

00:30:08.000 --> 00:30:13.000
the harder part of this problem.
It says let's take this matrix

00:30:13.000 --> 00:30:17.000
A and let's replace the two in
the upper right corner by some

00:30:17.000 --> 00:30:25.000
other number C.
That means we will look at 1 3

00:30:25.000 --> 00:30:34.000
C; 2 0 - 1; 1 1 0|.
And let's call that M.

00:30:34.000 --> 00:30:40.000
And it first asks you to find
the value of C for which this

00:30:40.000 --> 00:30:48.000
matrix is not invertible.
M is not invertible exactly

00:30:48.000 --> 00:30:53.000
when the determinant of M is
zero.

00:30:53.000 --> 00:31:05.000
Let's compute the determinant.
Well, we should do one times

00:31:05.000 --> 00:31:12.000
that smaller determinant,
which is zero minus negative

00:31:12.000 --> 00:31:16.000
one,
which is 1 times 1 minus three

00:31:16.000 --> 00:31:23.000
times that determinant,
which is zero plus one is 1.

00:31:23.000 --> 00:31:28.000
And then we have plus C times
the lower left determinant which

00:31:28.000 --> 00:31:31.000
is two times one minus zero is
2.

00:31:31.000 --> 00:31:39.000
That gives us one minus three
is - 2 2C.

00:31:39.000 --> 00:31:47.000
That is zero when C equals 1.
For C equals 1,

00:31:47.000 --> 00:31:53.000
this matrix is not invertible.
For other values it is

00:31:53.000 --> 00:31:58.000
invertible.
It goes on to say let's look at

00:31:58.000 --> 00:32:04.000
this value of C and let's look
at the system Mx equals zero.

00:32:04.000 --> 00:32:14.000
I am going to put value one in
there.

00:32:14.000 --> 00:32:20.000
Now, if we look at Mx equals
zero, well, this has either no

00:32:20.000 --> 00:32:24.000
solution or infinitely many
solutions.

00:32:24.000 --> 00:32:26.000
But here there is an obvious
solution.

00:32:26.000 --> 00:32:30.000
Namely x equals zero is a
solution.

00:32:30.000 --> 00:32:35.000
Maybe let me rewrite it more
geometrically.

00:32:35.000 --> 00:32:44.000
X 3 y z = 0.
2x - z = 0.

00:32:44.000 --> 00:32:52.000
And x y = 0.
You see we have an obvious

00:32:52.000 --> 00:32:54.000
solution, (0,0,
0).

00:32:54.000 --> 00:32:57.000
But we have more solutions.
How do we find more solutions?

00:32:57.000 --> 00:33:01.000
Well, (x, y,
z) is a solution if it is in

00:33:01.000 --> 00:33:06.000
all three of these planes.
That is a way to think about it.

00:33:06.000 --> 00:33:13.000
Probably we are actually in
this situation where,

00:33:13.000 --> 00:33:20.000
in fact, we have three planes
that are all passing through the

00:33:20.000 --> 00:33:25.000
origin and all parallel to the
same line.

00:33:25.000 --> 00:33:28.000
And so that would be the line
of solutions.

00:33:28.000 --> 00:33:32.000
To find it actually we can
think of this as follows.

00:33:32.000 --> 00:33:37.000
The first observation is that
actually in this situation we

00:33:37.000 --> 00:33:41.000
don't need all three equations.
The fact that the system has

00:33:41.000 --> 00:33:45.000
infinitely many solutions means
that actually one of the

00:33:45.000 --> 00:33:49.000
equations is redundant.
If you look at it long enough

00:33:49.000 --> 00:33:51.000
you will see,
for example,

00:33:51.000 --> 00:33:55.000
if you multiply three times
this equation and you subtract

00:33:55.000 --> 00:33:58.000
that one then you will get the
first equation.

00:33:58.000 --> 00:34:05.000
Three times (x y) - (2x - z)
will be x 3y z.

00:34:05.000 --> 00:34:08.000
Now, we don't actually need to
see that to solve a problem.

00:34:08.000 --> 00:34:10.000
I am just showing you that is
what happens when you have a

00:34:10.000 --> 00:34:14.000
matrix with determinant zero.
One of the equations is somehow

00:34:14.000 --> 00:34:19.000
a duplicate of the others.
We don't actually need to

00:34:19.000 --> 00:34:24.000
figure out how exactly.
What that means is really we

00:34:24.000 --> 00:34:28.000
want to solve,
let's say start with two of the

00:34:28.000 --> 00:34:33.000
equations.
To find the solution we can

00:34:33.000 --> 00:34:41.000
observe that the first equation
says actually that <x,

00:34:41.000 --> 00:34:45.000
y, z> dot-product with

00:34:46.000 --> 00:34:51.000
=0.
And the second equation says

00:34:52.000 --> 00:34:57.000
dot-product with <2,0,-
1> is zero.

00:34:57.000 --> 00:35:03.000
And the third equation,
if we really want to keep it,

00:35:03.000 --> 00:35:08.000
says we should be also having
this.

00:35:08.000 --> 00:35:11.000
Now, these equations now
written like this,

00:35:11.000 --> 00:35:15.000
they are just saying we want an
x, y, z that is perpendicular to

00:35:15.000 --> 00:35:19.000
these vectors.
Let's forget this one and let's

00:35:19.000 --> 00:35:23.000
just look at these two.
They are saying we want a

00:35:23.000 --> 00:35:27.000
vector that is perpendicular to
these two given vectors.

00:35:27.000 --> 00:35:35.000
How do we find that?
We do the cross-product.

00:35:35.000 --> 00:35:43.000
To find x, y,
z perpendicular to <1,3,

00:35:43.000 --> 00:35:51.000
1> and <2,0,
- 1>, we take the

00:35:51.000 --> 00:35:56.000
cross-product.
And that will give us

00:35:56.000 --> 00:35:58.000
something.
Well, let me just give you the

00:35:58.000 --> 00:36:00.000
answer.
I am sure you know how to do

00:36:00.000 --> 00:36:07.000
cross-products by now.
I don't have the answer here,

00:36:07.000 --> 00:36:18.000
so I guess I have to do it.
That should be <- 3,

00:36:18.000 --> 00:36:26.000
probably positive 3,
and then - 6>.

00:36:26.000 --> 00:36:29.000
That is the solution.
And any multiple of that is a

00:36:29.000 --> 00:36:31.000
solution.
If you like to neatly simplify

00:36:31.000 --> 00:36:34.000
them you could say negative one,
one, negative two.

00:36:34.000 --> 00:36:37.000
If you like larger numbers you
can multiply that by a million.

00:36:37.000 --> 00:36:50.000
That is also a solution.
Any questions about that?

00:36:50.000 --> 00:36:56.000
Yes?
That is correct.

00:36:56.000 --> 00:37:00.000
If you pick these two guys
instead, you will get the same

00:37:00.000 --> 00:37:02.000
solution.
Well, up to a multiple.

00:37:02.000 --> 00:37:06.000
It could be if you do the
cross-product of these two guys

00:37:06.000 --> 00:37:10.000
you actually get something that
is a multiple -- Actually,

00:37:10.000 --> 00:37:14.000
I think if you do the
cross-product of the first and

00:37:14.000 --> 00:37:17.000
third one you will get actually
minus one, one,

00:37:17.000 --> 00:37:20.000
minus two, the smaller one.
But it doesn't matter.

00:37:20.000 --> 00:37:22.000
I mean it is really in the same
direction.

00:37:22.000 --> 00:37:27.000
This is all because a plane has
actually normal vectors of all

00:37:27.000 --> 00:37:32.000
sizes.
Yes?

00:37:32.000 --> 00:37:35.000
I don't think so because -- An
important thing to remember

00:37:35.000 --> 00:37:38.000
about cross-product is we
compute for minors,

00:37:38.000 --> 00:37:40.000
but then we put a minus sign on
the second component.

00:37:40.000 --> 00:37:44.000
The coefficient of j in here,
the second component,

00:37:44.000 --> 00:37:47.000
you do one times minus two
times one.

00:37:47.000 --> 00:37:51.000
That is negative three indeed.
But then you actually change

00:37:51.000 --> 00:37:54.000
that to a positive three.
Yes?

00:38:14.000 --> 00:38:21.000
Well, we don't have parametric
equations here.

00:38:21.000 --> 00:38:25.000
Oh, solving by elimination.
Well, if it says that you have

00:38:25.000 --> 00:38:29.000
to use vector methods then you
should use vector methods.

00:38:29.000 --> 00:38:32.000
If it says you should use
vectors and matrices then you

00:38:32.000 --> 00:38:41.000
are expected to do it that way.
Yes?

00:38:41.000 --> 00:38:43.000
It depends what the problem is
asking.

00:38:43.000 --> 00:38:45.000
The question is,
is it enough to find the

00:38:45.000 --> 00:38:47.000
components of a vector or do we
have to find the equation of a

00:38:47.000 --> 00:38:50.000
line?
Here it says find one solution

00:38:50.000 --> 00:38:55.000
using vector operations.
We have found one solution.

00:38:55.000 --> 00:38:58.000
If you wanted to find the line
then it would all the things

00:38:58.000 --> 00:39:04.000
that are proportional to this.
It would be maybe minus 3t,

00:39:04.000 --> 00:39:09.000
3t minus 6t,
all the multiples of that

00:39:09.000 --> 00:39:12.000
vector.
We do because (0,0,

00:39:12.000 --> 00:39:17.000
0) is an obvious solution.
Maybe I should write that on

00:39:17.000 --> 00:39:19.000
the board.
You had another question?

00:39:28.000 --> 00:39:31.000
Not quite.
Let me re-explain first how we

00:39:31.000 --> 00:39:35.000
get all the solutions and why I
did that cross-product.

00:40:08.000 --> 00:40:09.000
First of all,
why did I take that

00:40:09.000 --> 00:40:12.000
cross-product again?
I took that cross-product

00:40:12.000 --> 00:40:17.000
because I looked at my three
equations and I observed that my

00:40:17.000 --> 00:40:21.000
three equations can be
reformulated in terms of these

00:40:21.000 --> 00:40:25.000
dot-products saying that x,
y, z is actually perpendicular

00:40:25.000 --> 00:40:29.000
these guys and these guys have
normal vectors to the planes.

00:40:29.000 --> 00:40:33.000
Remember, to be in all three
planes it has to be

00:40:33.000 --> 00:40:36.000
perpendicular to the normal
vectors.

00:40:36.000 --> 00:40:40.000
That is how we got here.
And now, if we want something

00:40:40.000 --> 00:40:43.000
that is perpendicular to a bunch
of given vectors,

00:40:43.000 --> 00:40:45.000
well, to be perpendicular to
two vectors,

00:40:45.000 --> 00:40:48.000
an easy way to find one is to
take that cross-product.

00:40:48.000 --> 00:40:51.000
And, if you take any two of
them, you will get something

00:40:51.000 --> 00:40:54.000
that is the same up to scaling.
Now, what it means

00:40:54.000 --> 00:41:00.000
geometrically is that when we
have our three planes and they

00:41:00.000 --> 00:41:06.000
all actually contain the same
line -- And we know that is

00:41:06.000 --> 00:41:11.000
actually the smae case because
they all pass through the

00:41:11.000 --> 00:41:16.000
origin.
They pass through the origin

00:41:16.000 --> 00:41:20.000
because the constant terms are
just zero.

00:41:20.000 --> 00:41:26.000
What happens is that the normal
vectors to these planes are,

00:41:26.000 --> 00:41:29.000
in fact, all perpendicular to
that line.

00:41:29.000 --> 00:41:40.000
The normal vectors -- Say this
line is vertical.

00:41:40.000 --> 00:41:45.000
The normal vectors are all
going to be horizontal.

00:41:45.000 --> 00:41:49.000
Well, it is kind of hard to
draw.

00:41:49.000 --> 00:41:53.000
By taking the cross-product
between two normal vectors we

00:41:53.000 --> 00:42:01.000
found this direction.
Now, to find actually all the

00:42:01.000 --> 00:42:06.000
solutions.
What we know so far is that we

00:42:06.000 --> 00:42:11.000
have this direction <-3 3 -
6>.

00:42:11.000 --> 00:42:14.000
That is going to be parallel to
the line of intersections.

00:42:14.000 --> 00:42:19.000
Let me do it here,
for example,

00:42:19.000 --> 00:42:29.000
.
Now we have one particular

00:42:29.000 --> 00:42:34.000
solution.
0,0, 0.

00:42:34.000 --> 00:42:39.000
Actually, we have found another
one, too, which is <- 3,3,

00:42:39.000 --> 00:42:46.000
- 6>.
Anyway, if a line of solutions

00:42:46.000 --> 00:42:55.000
-- -- has parametric equation x
= - 3t, y = 3t,

00:42:55.000 --> 00:43:04.000
z = - 6t, anything proportional
to that.

00:43:04.000 --> 00:43:07.000
That is how we would find all
the solutions if we wanted them.

00:43:18.000 --> 00:43:22.000
It is almost time.
I think I need to jump ahead to

00:43:22.000 --> 00:43:24.000
other problems.
Let's see.

00:43:24.000 --> 00:43:29.000
I think problem 4 you can
probably find for yourselves.

00:43:29.000 --> 00:43:32.000
It is a reasonably
straightforward parametric

00:43:32.000 --> 00:43:35.000
equation problem.
You just have to find the

00:43:35.000 --> 00:43:39.000
coordinates of point P.
And for that it is a very

00:43:39.000 --> 00:43:41.000
simple trick.
Problem 5.

00:43:41.000 --> 00:43:43.000
Find the area of a spaced
triangle.

00:43:43.000 --> 00:43:47.000
It sounds like a cross-product.
Find the equation of a plane

00:43:47.000 --> 00:43:50.000
also sounds like a
cross-product.

00:43:50.000 --> 00:43:53.000
And find the intersection of
this plane with a line means we

00:43:53.000 --> 00:43:56.000
find first the parametric
equation of the line and then we

00:43:56.000 --> 00:43:59.000
plug that into the equation of
the plane to get where they

00:43:59.000 --> 00:44:02.000
intersect.
Does that sound reasonable?

00:44:02.000 --> 00:44:06.000
Who is disparate about problem
5?

00:44:06.000 --> 00:44:15.000
OK. Let me repeat problem 5.
First part we need to find the

00:44:15.000 --> 00:44:19.000
area of a triangle.
And the way to do that is to

00:44:19.000 --> 00:44:22.000
just do one-half the length of a
cross-product.

00:44:22.000 --> 00:44:32.000
If we have three points,
P0, P1, P2 then maybe we can

00:44:32.000 --> 00:44:38.000
form vectors P0P1 and P0P2.
And, if we take that

00:44:38.000 --> 00:44:42.000
cross-product and take the
length of that and divide by

00:44:42.000 --> 00:44:46.000
two, that will give us the area
of a triangle.

00:44:46.000 --> 00:44:52.000
Here it turns out that this guy
is ,

00:44:52.000 --> 00:44:59.000
if I look at the solutions,
so you will end up with square

00:44:59.000 --> 00:45:06.000
root of 6 over 2.
The second is asking you for

00:45:06.000 --> 00:45:13.000
the equation of a plane
containing these three points.

00:45:13.000 --> 00:45:23.000
Well, first of all,
we know that a normal vector to

00:45:23.000 --> 00:45:34.000
the plane is going to be given
by this cross-product again.

00:45:34.000 --> 00:45:40.000
That means that the equation of
plane will be of a form x plus y

00:45:40.000 --> 00:45:45.000
plus 2z equals something.
If a coefficient is here it

00:45:45.000 --> 00:45:49.000
comes from the normal vector.
And to find what goes in the

00:45:49.000 --> 00:45:51.000
right-hand side,
we just plug in any of the

00:45:51.000 --> 00:45:55.000
points.
If you plug in P0,

00:45:55.000 --> 00:46:03.000
which is (2,1,
0) then two plus one seems like

00:46:03.000 --> 00:46:06.000
it is 3.
And, if you want to

00:46:06.000 --> 00:46:08.000
double-check your answer,
you can take P1 and P2 and

00:46:08.000 --> 00:46:15.000
check that you also get three.
It is a good way to check your

00:46:15.000 --> 00:46:18.000
answer.
Then the third part.

00:46:18.000 --> 00:46:25.000
We have a line parallel to the
vector v equals one,

00:46:25.000 --> 00:46:31.000
one, one through the point S,
which is (- 1,0,

00:46:31.000 --> 00:46:34.000
0).
That means you can find its

00:46:34.000 --> 00:46:37.000
parametric equation.
X will start at - 1,

00:46:37.000 --> 00:46:41.000
increases at rate 1.
Y starts at zero,

00:46:41.000 --> 00:46:44.000
increases at rate one.
Z starts at zero,

00:46:44.000 --> 00:46:48.000
increases at rate one.
You plug these into the plane

00:46:48.000 --> 00:46:52.000
equation, and that will tell you
where they intersect.

00:46:52.000 --> 00:46:58.000
Is that clear?
And now, in the last one

00:46:58.000 --> 00:47:03.000
minute,
on that side I have one minute,

00:47:03.000 --> 00:47:08.000
let me just say very quickly --
Well,

00:47:08.000 --> 00:47:10.000
do you want to hear about
problem 6 anyway very quickly?

00:47:10.000 --> 00:47:20.000
Yeah. OK.
Problem 6 is one of these like

00:47:20.000 --> 00:47:24.000
vector calculations.
It says we have a position

00:47:24.000 --> 00:47:28.000
vector R.
And it asks you how do we find

00:47:28.000 --> 00:47:32.000
the derivative of R dot R?
Well, remember we have a

00:47:32.000 --> 00:47:34.000
product rule for taking the
derivative.

00:47:34.000 --> 00:47:37.000
UV prime is U prime V plus UV
prime.

00:47:37.000 --> 00:47:44.000
It also applies for dot-product.
That is dR by dt dot R plus R

00:47:44.000 --> 00:47:49.000
dot dR by dt.
And these are both the same

00:47:49.000 --> 00:47:52.000
thing.
You get two R dot dR/dt,

00:47:52.000 --> 00:47:57.000
but dR/dt is v for velocity
vector.

00:47:57.000 --> 00:48:00.000
Hopefully you have seen things
like that.

00:48:00.000 --> 00:48:04.000
Now, it says show that if R has
constant length then they are

00:48:04.000 --> 00:48:10.000
perpendicular.
All you need to write basically

00:48:10.000 --> 00:48:15.000
is we assume length R is
constant.

00:48:15.000 --> 00:48:18.000
That is what it says,
R has constant length.

00:48:18.000 --> 00:48:21.000
Well, how do we get to,
say, something we probably want

00:48:21.000 --> 00:48:26.000
to reduce to that?
Well, if R is constant in

00:48:26.000 --> 00:48:32.000
length then R dot R is also
constant.

00:48:32.000 --> 00:48:39.000
And so that means d by dt of R
dot R is zero.

00:48:39.000 --> 00:48:41.000
That is what it means to be
constant.

00:48:41.000 --> 00:48:46.000
And so that means R dot v is
zero.

00:48:46.000 --> 00:48:52.000
That means R is perpendicular
to v.

00:48:52.000 --> 00:48:58.000
That is a proof.
It is not a scary proof.

00:48:58.000 --> 00:49:04.000
And then the last question of
the exam says let's continue to

00:49:04.000 --> 00:49:10.000
assume that R has constant
length, and let's try to find R

00:49:10.000 --> 00:49:13.000
dot v.
If there is acceleration then

00:49:13.000 --> 00:49:17.000
probably we should bring it in
somewhere, maybe by taking a

00:49:17.000 --> 00:49:21.000
derivative of something.
If we know that R dot v equals

00:49:21.000 --> 00:49:24.000
zero, let's take the derivative
of that.

00:49:24.000 --> 00:49:32.000
That is still zero.
But now, using the product

00:49:32.000 --> 00:49:40.000
rule, dR/dt is v dot v plus R
dot dv/dt is going to be zero.

00:49:40.000 --> 00:49:43.000
That means that you are asked
about R dot A.

00:49:43.000 --> 00:49:48.000
Well, that is equal to minus V
dot V.

00:49:48.000 --> 00:49:50.000
And that is it.