Vector Valued Functions in R3

Give an example of a vector-valued function r(t) and determine its domain and range in $R^3$.

Vector-valued functions are significant in the study of multivariable calculus because they allow us to describe a multitude of real-world phenomena, including motion, forces, and fields in three-dimensional space. A vector-valued function in $R^3$, such as r(t) = <f(t), g(t), h(t)>, maps a real number t in the domain of the function into a three-dimensional vector. The components f(t), g(t), and h(t) are real-valued functions of the parameter t, and can represent quantities like position as functions of time in physics.

When considering the domain of a vector-valued function r(t), we analyze the individual functions f(t), g(t), and h(t) to determine the largest set of values t for which all three functions are defined. These values collectively make up the domain of r(t). For instance, if all three component functions are defined for all real numbers, then the domain of r(t) is also all real numbers. Conversely, any restrictions on the domain of the individual components will restrict the domain of the vector function.

The range of a vector-valued function in $R^3$ is the set of all possible output vectors <f(t), g(t), h(t)> as t varies over its domain. The concept of range is crucial because it informs us about the spread of values that the function can take, which is particularly pertinent in applications such as determining the path of an object in space.

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