Voiceover:We have the vector A,
being equal to five, negative six.
If we wanted to visualize that,
we could start at the tail of vector A,
and we'd wanna move five,
positive five in the X directions.
That'd be five to the right based on our conventions.
We'd wanna move negative six in the Y directions,
and based on our conventions that would be six down.
This length would be six.
If this was a positive six,
we would have moved six up, over here.
This length over here is five.
Going five to the right,
and then six down would take us
from the tail to the head.
Now we are at the head,
or tip, or arrow side of the vector.
Let's draw it just like that.
That's a reasonable representation for our vector A.
What else can we think about?
How else could we think about this vector?
What if we wanted to figure out its magnitude?
The magnitude of this vector
is just the length of this arrow
if we think of it visually,
and this is a right triangle.
We just need to figure out the length
of the hypotenuse of this right triangle.
Well we know the lengths of the two shorter sides.
The magnitude of vector A
is just going to be equal to the square root.
This is coming straight out
of the Pythagorean theorem.
Square root of five squared.
Five squared plus six squared.
Which is equal to the square root of,
25 plus 36 is 61.
The length of this arrow, right over here,
is square root of 61,
which is a little bit under eight,
which actually just makes visual sense.
That was interesting.
Let's say we wanna do something else.
Let's say we wanted to figure out some unit vector
that goes in the same direction as A.
It's a unit vector.
That means that its length is one,
but it goes in the same direction as A.
We've seen this before.
If we call this unit vector U,
and I'll put the little hat on top of it
to show that it's a unit vector.
What we could do is, we could take vector A,
and divide it by, essentially scale it down,
by the magnitude, by the magnitude of vector A.
Another way of thinking about it,
this would be equal to.
This part right over here,
we could write it like this.
We could say one over the magnitude
of vector A, times vector A.
This makes it pretty clear
we're just scaling vector A.
This part, right over here,
that's going to be equal to one over square root of 61,
we already figured out that
this was already the square root of 61.
We're gonna multiply it times vector A,
which is five, negative six.
Five, negative 6.
Now we can scale it up, or scale it down,
in this case, by just multiplying
the scaler times each of these components.
U is equal to five over the square root of 61,
all I'm doing is I'm scaling each of the components
in the X direction and the Y direction
by one over the square root of 61,
negative six over the square root of 61.
That was interesting.
Let's ask ourselves another question.
What would we have to multiply U by to get A?
What scaler could we multiply it by to get A?
Our answer is actually right over here.
We see that the vector U is equal to the vector A,
essentially divided by this scaler.
If you multiply both sides times the scaler,
you get the magnitude of A,
which is just a number,
it's just a length of the side,
times our unit vector U,
is going to be equal to our original vector.
If we wanted to write our original vector,
kinda broken up this way,
we could say that our original vector,
is equal to its magnitude,
is equal to the square root of 61,
times our unit vector, that makes sense,
the length of this unit vector
is one square root of 61th of this entire length,
right over here, and this of course is going
to be the same thing as the square root
of 61 times all of this business.
Copy and paste times all that business,
and I think it might jump out at you
if you multiply the scaler,
you're gonna multiply it times each of these components,
you're going to be left with five, negative six.
The whole reason why I did this
is just to make you a little bit more comfortable
with the different ways
of manipulating these notations.
Sometimes when you see this type of thing
with these double lines on either side,
it looks like something very fancy,
but it's really just talking about the magnitude
of the vector, the length of the vector.
The whole point of this is just to relate
the vector itself to the magnitude
of the vector to a unit vector
going in the same direction.