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Hovedindhold

Aktuel tid:0:00Samlet varighed:5:49

A vector is something that has
both magnitude and direction. Magnitude and direction. So let's think of an example
of what wouldn't and what would be a vector. So if someone tells
you that something is moving at 5 miles per hour,
this information by itself is not a vector quantity. It's only specifying
a magnitude. We don't know what
direction this thing is moving 5 miles per hour in. So this right over
here, which is often referred to as a speed, is not a
vector quantity just by itself. This is considered to
be a scalar quantity. If we want it to be a
vector, we would also have to specify the direction. So for example,
someone might say it's moving 5 miles
per hour east. So let's say it's moving
5 miles per hour due east. So now this combined 5
miles per are due east, this is a vector quantity. And now we wouldn't
call it speed anymore. We would call it velocity. So velocity is a vector. We're specifying the
magnitude, 5 miles per hour, and the direction east. But how can we actually
visualize this? So let's say we're
operating in two dimensions. And what's neat
about linear algebra is obviously a lot
of what applies in two dimensions
will extend to three. And then even four, five, six,
as made dimensions as we want. Our brains have trouble
visualizing beyond three. But what's neat is
we can mathematically deal with beyond three
using linear algebra. And we'll see that
in future videos. But let's just go back to
our straight traditional two-dimensional vector
right over here. So one way we
could represent it, as an arrow that
is 5 units long. We'll assume that each of our
units here is miles per hour. And that's pointed
to the right, where we'll say the right is east. So for example, I could start
an arrow right over here. And I could make its length 5. The length of the arrow
specifies the magnitude. So 1, 2, 3, 4, 5. And then the direction
that the arrow is pointed in specifies
it's direction. So this right over here could
represent visually this vector. If we say that the
horizontal axis is say east, or the positive horizontal
direction is moving in the east, this would be
west, that would be north, and then that would be south. Now, what's interesting
about vectors is that we only care about the magnitude
in the direction. We don't necessarily
not care where we start, where we place it when we think
about it visually like this. So for example, this would
be the exact same vector, or be equivalent vector to this. This vector has the same length. So it has the same magnitude. It has a length of 5. And its direction
is also due east. So these two vectors
are equivalent. Now one thing that you might say
is, well, that's fair enough. But how do we represent
it with a little bit more mathematical notation? So we don't have to
draw it every time. And we could start
performing operations on it. Well, the typical
way, one, if you want a variable to
represent a vector, is usually a lowercase letter. If you're publishing a
book, you can bold it. But when you're doing
it in your notebook, you would typically put a
little arrow on top of it. And there are several
ways that you could do it. You could literally say,
hey 5 miles per hour east. But that doesn't feel
like you can really operate on that easily. The typical way is to specify,
if you're in two dimensions, to specify two
numbers that tell you how much is this vector moving
in each of these dimensions? So for example,
this one only moves in the horizontal dimension. And so we'll put our
horizontal dimension first. So you might call
this vector 5, 0. It's moving 5, positive 5
in the horizontal direction. And it's not moving at all
in the vertical direction. And the notation might change. You might also see notation, and
actually in the linear algebra context, it's more
typical to write it as a column vector
like this-- 5, 0. This once again,
the first coordinate represents how much we're moving
in the horizontal direction. And the second coordinate
represents how much are we moving in the
vertical direction. Now, this one isn't
that interesting. You could have other vectors. You could have a vector
that looks like this. Let's say it's moving 3 in
the horizontal direction. And positive 4. So 1, 2, 3, 4 in the
vertical direction. So it might look
something like this. So this could be another
vector right over here. Maybe we call this
vector, vector a. And once again, I want to
specify that is a vector. And you see here that if
you were to break it down, in the horizontal direction,
it's shifting three in the horizontal direction,
and it's shifting positive four in the vertical direction. And we get that by
literally thinking about how much we're moving
up and how much we're moving to the right when we
start at the end of the arrow and go to the front of it. So this vector might
be specified as 3, 4. 3, 4. And you could use the
Pythagorean theorem to figure out the actual
length of this vector. And you'll see because this is
a 3, 4, 5 triangle, that this actually has a magnitude of 5. And as we study more
and more linear algebra, we're going to start extending
these to multiple dimensions. Obviously we can visualize
up to three dimensions. In four dimensions it
becomes more abstract. And that's why this type
of a notation is useful. Because it's very hard to draw
a 4, 5, or 20 dimensional arrow like this.