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Previously we were talking about vectors and vectors, as we said, had a quantity
where to define it you needed a magnitude
and direction.
And we identified a vector with a symbol and a line over it.
and also quantities for a vector is that we could translate it anywhere we wanted
on our coordinate system or map as long as we didn't rotate the vector.
We could translate a vector because the starting point and the stopping point doesn't matter for a vector - independent
it only depends on the magnitude and the direction
and that's why you can't rotate a vector because that changes the direction that it's pointing.
We discussed how to multiply by a scalar, and add and subtract them,
and we also looked at how to represent a vector
as a pair of coordinates
called "components"
where, assuming that the tail of the vector was at the origin of the coordinate system,
the aX and aY were the X and Y position of the tip.
and so given those components we could calculate the magnitude
of the vector,
and the magnitude
of the vector could be represented by its absolute value
or possibly just the symbol itself without the line,
and it was equal to the square root of the sums of the components.
It's important to note that the magnitude here is a scalar quantity.
And so now we want to talk about a very special type of vector today which is the unit vector.
So what classifies a unit vector?
The characteristics of the unit vector are that it is: (1) dimensionless
so it does not have any units, it's not a length of meters or meters per second- it has to be dimensionless.
and the second quantity of a unit vector
is it has a magnitude of one.
So what does that mean exactly?
So, say if I had a unit vector then - unit vector
and I wanted to write its magnitude and direction, it's magnitude would always be one
and it would have some direction associated with it.
Because it's a special type of vector we have a special type of notation,
which is the symbol with a little carat over it, or a hat, so we call that "A-hat"
Now what makes a unit vector special
is that it gives you information
about the direction
of the vector
If you think about the magnitude here... the magnitude gives you information about the amount
of the vector, the quantity, how large it is, the magnitude,
whereas the unit vector always has a magnitude of one and is dimensionless
It only contains information about the direction of the vector.
And so that leads us to an interesting equation
which is that any vector
can be written as
can be written as
its magnitude times its unit vector
its magnitude times its unit vector.
and we'll just draw a little box around that because this is an important way to look at a vector.
where we've decomposed our vector into a quantity that characterizes,
that contains the information concerning the amount of the vector
and a quantity that contains the information just about the direction of the vector.
so let's look at an example. Let's say I have a vector "a"
and it has a magnitude
of 5 meters
of 5 meters
and it has a direction of - we'll give ourselves a nice triangle - 36.87°
counterclockwise from the positive X axis.
and so this can be re-written as a scalar
5 meters
times the vector
of magnitude one and dimensionless, with the direction 36.87° counterclockwise from the positive X axis
and so in this case here
this is the magnitude "a"
and this is the unit vector "a-hat"
that contains the direction information while the magnitude contains the information of the amount.
It's useful to turn this around for a minute and look at it and imagine you're solving for a-hat.
If you had a vector and you wanted to derive the unit vector
you can do that by simply taking the vector
and dividing by the magnitude
So for example if I do this and I'd just, 1/5m, that's one over the magnitude,
times the original vector, which is 5m, 36.87° counterclockwise X
If I then multiply this through, I multiply the scalar by the magnitude, the meters units cancel,
the fives cancel, I end up with 1, 36.87°.
Now while this is pretty easy, this same procedure
follows through in all the different representations we have for vectors
this becomes a way to calculate a unit vector if we have the regular vector first.
OK, now that we have an idea of what a unit vector is, I want to introduce
an even more very special unit vector,
OK so let's go back to our coordinate system
Let's give ourselves a coordinate system here
And so in a coordinate system I always want to identify my positive X and positive Y axes.
And this will work in three dimensions but we'll just stay in two dimensions for now.
and I'm going to call the vector "i-hat" to be a vector - it's a unit vector so it has magnitude one
and it points along the positive X axis
and so it would be a vector
that has a magnitude one and it points along the positive X.
My other vector, called "j-hat", also has a magnitude one
and it has a direction along the positive Y axis.
And so if this is "i-hat"
then "j-hat" points along the postive Y.
It's important that - we'll use these vectors over and over -
it's important to remember that they're just vectors.
They have special conditions: they have magnitude one and these particular ones have specific directions that they point,
but in every respect we can treat them like normal vectors
So if we were too represent them with coordinates, if we're to look at the coordinates
"i-hat" - it's tip is located at X=1 and Y=0
where "j-hat" then, if you were to find the components of "j- hat",
the tip is at X=0 and Y=1.
and so that is how we would write those same unit vectors just in terms of an ordered pair of their components
and so that is how we would write those same unit vectors just in terms of an ordered pair of their components.
and so that is how we would write those same unit vectors just in terms of an ordered pair of their components
so let's go back to this vector that we had before
this one here
if we were to put that on our graph
it would point
off in this direction and has some
length of 5 meters
and it had some angle
counterclockwise from the positive X axis
and so how would we represent that in component form?
Well, what we'd do here is
if we want to find the components of this we'd draw a triangle
and the first is we'd need to find the X coordinate of the tip, and that's right here.
and if we used our trig rules
we would find that the X coordinate of the tip of the vector is at X=4
We need to find the Y coordinate tip
and if we use our trigonometry rules we'd find that the Y coordinate
here is a 3 - this is a 3:4:5 triangle
which we knew from when we set it up that way
but if we didn't remember that we could calculate these
this is the hypotenuse times cosine theta, and this is the hypotenuse times sine theta.
So here's our vector "a" now in component form.
So let us take a look at another way we can represent this vector "a" in terms of these new very special unit vectors.
So imagine a vector along the X axis
that covers the base of this triangle
so that first vector - we'll call it a1 here -
so that first vector - we'll call it a1 here -
so that first vector - we'll call it a1 here -
so this a1 vector
is going to have a magnitude 4
and it points along the positive X axis
Well, so that is the the same as 4 times the unit vector along the positive X axis,
which is "i-hat"
so then if I want
to look at this second vector here, which I'll call a2
a2 has a magnitude of 3
it points along the positive Y axis
and so it's the same as a magnitude 3 times the unit vector along the positive Y axis which is "j-hat"
so since the sum of these two vectors
gives me my original vector "a"
I can rewrite my vector "a"
as 4 "i-hat" plus 3 "j-hat"
Now this is referred to as "writing the vector in component form"
and probably the most common way in which we work with vectors
but it's important to know that both of these are individual vectors.
We're writing this as the sum of two vectors - one along the X direction and one along the Y.
These - the 4 and the 3 are the components
the X and Y components
and then
the "i" and the "j" are the unit vectors along the X and Y axis
and so this is component form.
And so this is how we use unit vectors to be able to write the new way of working with vectors
and in the next segment we'll learn how to add the vectors in component form.