What is the connection between vector functions and space curves?

Question

Virginia Palmer · Accepted Answer

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))

What is the connection between vector functions and space curves?

Answered question

Answer & Explanation

New Questions in Linear algebra