Welcome to introduction
to Vector Valued Functions.
The goals of this video are
to define a vector valued function
and also to graph a vector valued function.
A vector valued function,
sometimes called a vector function,
is given by a function in this form.
The vector valued function r of t is equal to
f of t times the unit vector i,
plus g of t times the unit vector j,
and we can also express this in component form.
And this would be for a 2D curve or a plane curve.
Now what I want you to notice here
is that for a given value of t,
the vector valued function returns a vector,
and function f and function g for a given value of t,
would return a real numerical value.
And for a vector valued function in R3 or 3D,
it would be expressed in a similar way
except now we have the z component
expressed by h of t times the unit vector k,
or in component form as we see here.
The curves produced by vector valued functions
are traced out by the terminal points
of the vectors formed by the function values
of f, g, and h.
To get a better idea of this,
let's take a look at a couple graphs.
Here we see a plane curve in blue,
and in red, we're going to see the vectors
that trace out this curve from the vector valued function.
Here we're starting at t equals negative three,
and watch as t increases.
The terminal point of this red vector
traces out the blue curve
as we see here.
So the terminal points of the vectors
produced by the vector valued function creates a curve.
Now let's take a look at a space curve
generated by a vector valued function.
Again, here we see the vector produced
by the vector valued function in red,
and the curve the terminal points produce
as the value of t changes.
Here we have t equals negative eight,
and as it increases to positive eight,
we see that it traces out the blue curve.
One other thing we'll notice about vector valued function
is if the curves are traced out
in a specific direction called the orientation,
and a lot of times when we're asked to graph
a vector valued function, we need to show
the orientation by placing arrows on the curve.
Let's take a look at a couple of examples.
Here we want to evaluate the vector valued function
for each value of t.
Then we want to write an equation for the function
in rectangular form,
and then graph the vector valued function.
So for r of zero, we replace t with zero.
So we'd have zero times the i vector,
plus negative zero squared
plus five
times the j vector.
So notice here, we have zero times the i vector,
plus five times vector j.
So we could express this as five j,
or if you want to express it in component form,
it would be zero, five.
This is one vector generated by the vector valued function
where the terminal point of this vector
would be one point on the plane curve.
So for r of two, we would have two times i,
plus negative two squared,
plus five, times vector j.
And notice, we're not squaring the negative two,
just the two, so here, we're going to have two i.
And this is going to be plus negative four plus five,
or one j.
Or again in component form, it would be two, one.
And for r of negative three, we would have negative three
times vector i, plus, here we'll have negative
negative three squared plus five times j.
That's going to give us negative three i,
and here we're going to have negative nine plus five,
that'll be minus four j, or in component form,
we'd have the vector negative three, negative four.
Okay, for part two, we want to write this equation
in rectangular form.
So what we'll first do is write the equation
in parametric form, and then eliminate the parameter t,
leaving an equation in terms of only Xs and Ys.
Well, here we have the x component
and here we have the y component.
So x would be equal to t.
And for the y component, we have y equals
negative t squared plus five.
Well, if we need an equation in terms of x and y,
and x is equal to t, we can perform substitution
into the second equation, replacing t with x.
So an equation for this curve in rectangular form
would be y equals negative x squared plus five.
And the last part, graph the vector valued function,
we'll go ahead and use software for that.
We saw it earlier.
Here it is again.
And again, notice that as t increases,
the terminal points of the red vectors
produced by the vector valued function
trace out the blue curve.
And again, notice the orientation is from left to right
as t increases.
Let's go and take a look at one more example.
Here we have a space curve because it has an x, y,
and z component.
We want to evaluate the vector valued function
for each value of t again, and then graph the function.
And again, the main thing to remember here
is the output of a vector valued function will be a vector.
So we'll have two times cosine zero times vector i,
plus two times sine zero, times vector j,
plus zero times vector k.
Well, the cosine of zero degrees would be one,
so we have two i, the sine of zero is zero,
so we'd have zero j and zero k, or in component form,
we would have two, zero, zero.
You normally don't have to express it both ways,
but I am just to emphasize the two ways.
Next, we have t equals pi over four.
Remember, the cosine of pi over four
and the sine of pi over four would be the same.
it's going to be square root two over two.
Well, two times square root two over two,
the twos would simplify, so we're going to have
the square root of two i, and the same
for two times sine of pi over four,
that's going to be square root two j,
and this will be pi over four k.
Or again in component form, we would have
square root of two, square root of two,
and pi over four.
Okay, and lastly, we have r of two pi over three.
And just to review, let's go ahead
and sketch the reference triangle for two pi over three.
It's going to be a 60 degree reference angle
in the second quadrant.
So for the cosine of two pi over three,
we're going to have negative 1/2.
For the sine of two pi over three,
we'll have square root three over two.
So let's go ahead and simplify this.
We're going to have negative i plus square root three j,
plus two pi over three k,
or in component form,
we would have negative one, square root three,
and two pi over three.
And we saw the graph of this earlier.
Let's go ahead and take a look again.
So here we see the vector produced
by the vector valued function,
and when t is equal to negative eight.
And watch as t increases,
it'll trace out the space curve,
and notice the orientation.
So when we evaluate that vector valued function
for different values of t, we were finding
individual vectors that trace out this curve.
Okay, that's going to do it for this introductory video.
I hope this gave you a good feel
for what vector valued functions are.