Vector-Valued Functions
The output of a vector-valued function is a vector. The vector function is sometimes referred to as vector–valued function. In many books,
vector function is denoted by         or
r in bold. When writing, it is difficult to write “r" as bold, so we write it as,       . Vector is a quantity, which
has both magnitude (length) and direction. Magnitude and direction is defined by components of a vector.
Mian Systems LLC
Vector-Valued Functions or Vector Function
Vectors are made of components. When you break a vector into parts, the parts are called components.
Each part of a two-dimensional vector, say <4, 5> corresponds to the x-axis (horizontal) component is 4,
and the y-axis (vertical) component is 5. This means your path is towards 4 on x-axis and towards 5, up
(vertical) on y-axis.  In the two dimensional figure below, the vector “v” in the component form is written as
v = <vx , vy> => v = <4.5>.
For example, if we draw a t axis with ‘a’, ‘b’ end points and at some instance we have t0; that is the place of the fly at that moment. Then
there is a vector corresponding to that particular time t0.
We give this function a name--          and we show the various paths of the fly at particular time through vector        (ta)--start point,        
(t0)--snapshot point and        (b)--end point.       (t0) is pronounced as
r of t0.

In the picture below, fly starts at end point t(a), changes direction and stops at t(b). On the flight path, we have a snap shot at t(0). The curve
is C and can be measured for its length.
Typically, a physics problem gives you an angle and a magnitude to define a vector and you have to find
the components of vector using trigonometry. In the case of a ball rolling on a table, it can roll at an angle
θ and the magnitude will be the speed at which the ball is rolling.
In some books the vector function is given by the following equation:
Source: http://www.youtube.com/watch?v=5ymFpq7PPxM
Click to read more....
i and j are unit vectors
Equation is parametric form and rectangular form?
Further analysis of a vector function
We hope that this tutorial moves faster than watching a video tutorial. Where credit for the material used in this tutorial
is due, we have provided. In a few places some of the images shown below are not our original work. We have
misplaced the credits as we were building this web page. However, we have made notification on these images to
explain the concept further.
Vector function gives you a history on a path of a curve.  Think of a fly trained to take a certain path, which is the curve. You can trace
through vectors where the fly was at a particular time. If you have two flies, you will have two histories on the same curve. Vectors are
snap shots of a particular time.