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- [Instructor] In other
videos, we have talked

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about how a vector can
be completely defined

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by a magnitude and a
direction, you need both.

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And here we have done that.

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We have said that the magnitude

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of vector a is equal to three units,

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these parallel lines here on both sides,

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it looks like a double absolute value.

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That means the magnitude of vector a.

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And you can also specify
that visually by making sure

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

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And we also have its direction.

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We see the direction of
vector a is 30 degrees

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counter-clockwise of due East.

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

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or another way to specify
or to define a vector.

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And that's by using components.

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

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we're gonna think about the tail

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of this vector and the
head of this vector.

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

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what is our change in x?

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And we could see our change

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in x would be that right over there.

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

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

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

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

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So let me label these.

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

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And if you think about it,

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

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you could reconstruct this
vector right over here

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by starting here, having that change in x,

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

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of the vector would be
relative to the tail.

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The notation for this is
we would say that vector a

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is equal to, and we'll have parenthesis,

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

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And so if we wanted to get tangible

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for this particular
vector right over here,

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we know the length of
this vector is three.

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Its magnitude is three.

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

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and then this is going
straight up and down.

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This is a right triangle.

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

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

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but we could use a little bit of geometry,

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or a little bit of
trigonometry to establish,

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if we know this angle,
if we know the length

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of this hypotenuse, that
this side that's opposite

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the 30 degree angle is gonna
be half the hypotenuse,

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so it's going to be 3/2.

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And that the change in x is going to be

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the square root of three times the 3/2.

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

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And so up here, we would
write our x component

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is three times the square
root of three over two.

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

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Now I know a lot of you might be thinking

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this looks a lot like coordinates
in the coordinate plane,

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where this would be the x coordinate

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and this would be the y coordinate.

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But when you're dealing with vectors,

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that's not exactly the interpretation.

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It is the case that if the vector's tail

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were at the origin right
over here, then its head

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would be at these coordinates
on the coordinate plane.

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But we know that a vector is not defined

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by its position, by the
position of the tail.

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I could shift this vector around wherever

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and it would still be the same vector.

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It can start wherever.

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So when you use this
notation in a vector context,

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these aren't x coordinates
and y coordinates.

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

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Let me do one more example to show

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that we can actually go the other way.

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So let's say I defined some vector b,

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

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

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

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So it would, if this is its tail,

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and its x component which is its change

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in x is square root of two.

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So it might look something like this.

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

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

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So I could write our change in y over here

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is square root of two.

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And so the vector would
look something like this.

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

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and we can use a little bit of geometry

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to figure out the magnitude

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and the direction of this vector.

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You can use the Pythagorean
theorem to establish

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that this squared plus this squared

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is gonna be equal to that squared.

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

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having a length of two, which tells you

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that the magnitude of
vector b is equal to two.

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

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you could do a little bit of trigonometry

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or even a little bit
of geometry recognizing

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

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

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So these are gonna be the same angles

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which are gonna be 45 degree angles.

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

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45 degrees counter-clockwise of due East.

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So hopefully you appreciate
that these are equivalent ways

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of representing a vector.

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You either can have a
magnitude and a direction,

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or you can have your components

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and you can go back and
forth between the two.

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