Curve Described by a VectorValued Function

Given the vector-valued function $r(t) = \langle 4\cos(t), 4\sin(t), 3 \rangle$, determine the curve it describes in 3D space.

In this problem, we are given a vector-valued function that describes a curve in three-dimensional space. The function is defined in terms of trigonometric components involving cosine and sine functions, suggesting a circular or elliptic path in the plane that these two components define. The constant third component implies that the path is traced out parallel to a particular plane, maintaining a fixed height or depth.

The task here is to identify and determine the nature of the curve traced by this function. The presence of $4\cos(t)$ and $4\sin(t)$ as the first two components suggests that in the xy-plane, the path is a circle of radius 4 centered at the origin. This is because these components fully describe the parametric equations of a circle where the x and y values oscillate sinusoidally and cosinusoidally, tracing out a complete circle as the parameter t varies.

In three-dimensional space, the presence of a constant z-component, which is 3 in this case, implies that the circle is not in the xy-plane, but rather parallel to it, elevated or depressed by a constant amount. Thus, the curve described is a circle belonging to a plane parallel to the xy-plane at z=3. This problem reinforces understanding of how altering vector-valued functions affects the spatial curve they trace, emphasizing concepts of parametric equations and their geometric interpretations in vector calculus.

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