Describing a 3D Parametric Curve

Given the vector-valued function $r(t) = \langle 0, 3\cos(t), 5\sin(t) \rangle$, describe the curve in 3D space.

When examining a vector-valued function like $r(t) = \langle 0, 3\cos(t), 5\sin(t) \rangle$, we're diving into the world of parametric curves in three-dimensional space. The primary goal here is to interpret how the given function describes a path or curve, and what that path looks like geometrically.

This particular function is intriguing because it separates the dependency of the coordinates on the parameter t, and it becomes obvious that while x-coordinate remains constant at zero, y and z vary with trigonometric functions. This implies that the curve is confined to a certain plane, specifically the yz-plane in this context. By examining the components, this curve traces out an ellipse because both y and z are described using sine and cosine functions, which are characteristic of circular motion when squared and added as per Pythagoras’ identity.

Interpreting vector-valued functions involves recognizing these kinds of geometric relationships and understanding how the functions describe motion or sets of points in space. It requires an integrated knowledge of trigonometric identities, visualization in multiple dimensions, and an understanding of the planes in which such curves exist. Gaining comfort with these interpretations is key in fields such as physics and engineering where modeling of paths and spaces comes into play regularly.

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