A space curve is a curve in space. There is a close connection between space
curves and vector functions. Specitically, we can determine a vector fiunction
which traces along a space curve C (provided we put the tail of the vectors at
the origin, so they are position vectors). Likewise, any vector function defines
a space curve.

This can be described as follows:

Suppose C 1s a curve in space. Then we can determine parametric equations for C (equations which tell us the coordinates of a particle travelling along at a given time t). Suppose x = f(t), y = g(t) and z = h(t) are parametric equations for C

Define a vector function r(t) = f(t)i+g(t)j+h(t)k which we shall call a vector function of C. We claim that as t varies, the position vector r(t) traces out the curve C

To see this, observe that any point on C has coordinates (f(t), g(t), h(t)), and any position vector r(t) has head at the point (f(t), g(t), h(t))

This can be described as follows:

Suppose C 1s a curve in space. Then we can determine parametric equations for C (equations which tell us the coordinates of a particle travelling along at a given time t). Suppose x = f(t), y = g(t) and z = h(t) are parametric equations for C

Define a vector function r(t) = f(t)i+g(t)j+h(t)k which we shall call a vector function of C. We claim that as t varies, the position vector r(t) traces out the curve C

To see this, observe that any point on C has coordinates (f(t), g(t), h(t)), and any position vector r(t) has head at the point (f(t), g(t), h(t))