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: