In most parts of this chapter, we will be dealing with vector functions of one variable. It is vital that the student gets acquainted which such functions as a sound knowledge is required when dealing with vector functions of several variables in the vector integral calculus A vector function takes the form:

Such a function is referred to as a vector function of one real variable where the variable concern is t. It takes a scalar real value and is sometimes termed parameter. The vector  is of course a vector, having both direction and magnitude. For example, we might have,

For various values of t, we’ll get different vectors, i.e.,  and .

The functions  are the component functions, or components, of . We call  continuous if each component function is continuous. For example,  is continuous for all t while  is continuous for t > 0.

We say that  is differentiable if each component function is differentiable. If so, we define the first derivative as

Note that by differentiating a vector function, we get another vector function. As with most real value functions, a vector function may not be differentiable. Consider

While  is continuous for all t, we can’t differentiate it because one of its component functions  is not differentiable and so  is not differentiable.

Now that you got a rough feel of vector functions, here comes the most important part. You need to divorce, well at least the geometrical aspect, the vector functions from the real-value functions in term of its representation on a graph. See, for a real-value function, you pick a value  for x, get a corresponding for y and plot the point on the graph. For vectors, the vector function gives you a VECTOR, an arrow in the space which brings you from a point to another point. The starting point is the origin and the ending point is the point where the vector brings you. Using different values of the parameter gives you different vectors.

And so, geometrically, we may envision a vector function as an adjustable arrow pivoted at the origin.

 represents the arrow from the origin to the point  and so for different values of t, we get different vectors of  . As t varies, the arrow sweeps out a curve in space. This curve is the locus of points .

At first sight, it seems that the vector function does give us a curve. Yes ultimately it does, and the term curve will also be used in vector calculus. But it is vitally important to know that this curve is traced out by vectors from the vector function. Sometimes this curve is called a trajectory and the vector  is called the position vector of this curve.

So remember, a curve in vectors are the points which vectors from vector function travel to from the origin.