What is the connection between vector functions and space curves?

Marla Payton 2022-01-05 Answered
What is the connection between vector functions and space curves?

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Virginia Palmer
Answered 2022-01-06 Author has 1123 answers
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))
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