Introduction to vector components | Vectors | Precalculus | Khan Academy

- [Instructor] In other
videos, we have talked

about how a vector can
be completely defined

by a magnitude and a
direction, you need both.

And here we have done that.

We have said that the magnitude

of vector a is equal to three units,

these parallel lines here on both sides,

it looks like a double absolute value.

That means the magnitude of vector a.

And you can also specify
that visually by making sure

that the length of this vector
arrow is three units long.

And we also have its direction.

We see the direction of
vector a is 30 degrees

counter-clockwise of due East.

Now in this video, we're
gonna talk about other ways

or another way to specify
or to define a vector.

And that's by using components.

And the way that we're gonna do it is,

we're gonna think about the tail

of this vector and the
head of this vector.

And think about as we go
from the tail to the head,

what is our change in x?

And we could see our change

in x would be that right over there.

We're going from this x
value to this x value.

And then what is going
to be our change in y.

And if we're going from
down here to up here,

our change in y, we can
also specify like that.

So let me label these.

This is my change in x, and
then this is my change in y.

And if you think about it,

if someone told you your
change in x and change in y,

you could reconstruct this
vector right over here

by starting here, having that change in x,

then having the change in y
and then defining where the tip

of the vector would be
relative to the tail.

The notation for this is
we would say that vector a

is equal to, and we'll have parenthesis,

and we'll have our change
in x comma, change in y.

And so if we wanted to get tangible

for this particular
vector right over here,

we know the length of
this vector is three.

Its magnitude is three.

We know that this is, since
this is going due horizontally

and then this is going
straight up and down.

This is a right triangle.

And so we can use a little
bit of geometry from the past.

Don't worry if you need a little
bit of a refresher on this,

but we could use a little bit of geometry,

or a little bit of
trigonometry to establish,

if we know this angle,
if we know the length

of this hypotenuse, that
this side that's opposite

the 30 degree angle is gonna
be half the hypotenuse,

so it's going to be 3/2.

And that the change in x is going to be

the square root of three times the 3/2.

So it's going to be three,
square roots of three over two.

And so up here, we would
write our x component

is three times the square
root of three over two.

And we would write that
the y component is 3/2.

Now I know a lot of you might be thinking

this looks a lot like coordinates
in the coordinate plane,

where this would be the x coordinate

and this would be the y coordinate.

But when you're dealing with vectors,

that's not exactly the interpretation.

It is the case that if the vector's tail

were at the origin right
over here, then its head

would be at these coordinates
on the coordinate plane.

But we know that a vector is not defined

by its position, by the
position of the tail.

I could shift this vector around wherever

and it would still be the same vector.

It can start wherever.

So when you use this
notation in a vector context,

these aren't x coordinates
and y coordinates.

This is our change in x,
and this is our change in y.

Let me do one more example to show

that we can actually go the other way.

So let's say I defined some vector b,

and let's say that its x
component is square root of two.

And let's say that its y
component is square root of two.

So let's think about what
that vector would look like.

So it would, if this is its tail,

and its x component which is its change

in x is square root of two.

So it might look something like this.

So that would be change in x
is equal to square root of two.

And then its y component would
also be square root of two.

So I could write our change in y over here

is square root of two.

And so the vector would
look something like this.

It would start here and
then it would go over here,

and we can use a little bit of geometry

to figure out the magnitude

and the direction of this vector.

You can use the Pythagorean
theorem to establish

that this squared plus this squared

is gonna be equal to that squared.

And if you do that,
you're going to get this

having a length of two, which tells you

that the magnitude of
vector b is equal to two.

And if you wanted to figure
out this angle right over here,

you could do a little bit of trigonometry

or even a little bit
of geometry recognizing

that this is going to be a
right angle right over here,

and that this side and that
side have the same length.

So these are gonna be the same angles

which are gonna be 45 degree angles.

And so just like that, you could
also specify the direction,

45 degrees counter-clockwise of due East.

So hopefully you appreciate
that these are equivalent ways

of representing a vector.

You either can have a
magnitude and a direction,

or you can have your components

and you can go back and
forth between the two.

And we'll get more practice
of that in future videos.