Suppose a frog is fired from a cannon at ground level, at an angle of 60 degrees
to the ground, with an intial velocity of `100` ft/s. Suppose we would like to
represent the position of the frog by a vector function, `vec p= < x, y >`, where
`x` represents the horizontal distance the frog has traveled and `y` represents
the height of the frog. Both `x` and `y` are functions of time, independent of
We can break down the inital velocity into its initial horizontal and its initial
vertical components, using trigonometry. By special right triangles, we see that
the initial horizontal velocity is `50` ft/s and the initial vertical velocity
is `50 sqrt(3)` ft/s.
Galileo discovered that horizontal velocity of a projectile remains constant
(ignoring air resistance, at least). Thus, the horizontal speed of the frog
remains at a constant `50` ft/s for his entire flight. Thus, the horizontal
distance the frog has traveled at time, `t` is given by `x=50t`.
The horizontal velocity of the frog, no doubt, changes during his flight. This is
because of acceleration due to gravity. From calculus, one can derive the frog's
height, `h(t)` at time `t` to be `h(t)=50 sqrt(3)t-32t^2`.
We now know the frog's horizontal position in terms of time and his height in
terms of time. Therefore, `vec p= < 50t, 50 sqrt(3)t-32t^2 >` is the vector
function that outputs a vector which points from the origin to the frog's position
at time `t`.
There are some functions that are difficult to represent in `y=f(x)` form, such as
the unit circle. It must be split into two separate functions to even be represented
in `y=f(x)` form, since it is not a function. This is a good example of a funciton
that is much easier to represent in parametric form. In this example, of course, we
will look at it in vector function form.
If we let `vec v= < x(t), y(t) >` where `x(t)=cos(t)` and `y(t)=sin(t)`, we get
a vector function, whose vectors point to points on the unit circle. By plugging in
various values of `t`, you can see that as `t` increases, the vector moves counter-clockwise
around the circle. In the example to the right, `t` is between `0` and `2 pi`. But since
since `cos(t)` and `sin(t)` are cyclical, the vector function repeatedly draws out
the unit circle for all values of `t`.