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Unformatted text preview: Section 14.1 Vector Functions and Space Curves Functions whose range does not consists of numbers A bulk of elementary mathematics involves the study of functions - rules that assign to a given input a particular output. In nearly all mathematics, these functions have had as their inputs and outputs real numbers (like f ( x ) = x 2 ). In this section, we introduce the idea of a vector function - a function whose outputs are vectors. 1. Vector Function Basics We start with the formal definition of a vector function. Definition 1.1. A vector-valued function, or a vector function, is a function whose domain is a set of real numbers and whose range is a set of vectors. We have already seen lots of examples of vector functions. Example 1.2. Let vector r ( t ) = (2 + 2 t ) vector i + (2 + 2 t ) vector j + (2- t ) vector k. Recall that this is a vector equation for a line which passes through the point (2 , 2 , 2) and points in the direction of 2 vector i + 2 vector j- vector k . It is also a vector function with independent variable t . In general, a vector function in 3-space can be written in component form just like equations for lines i.e. any vector function is of the form vector r ( t ) = f ( t ) vector i + g ( t ) vector j + h ( t ) vector k where f ( t ), g ( t ), and h ( t ) are scalar functions of t . We call these function the component functions of the vector function vector r ( t ). As with regular functions, the usual definitions apply such as domain , range etc. We can also define other notions such as limits as continuity in of a vector function in terms of the component functions. Definition 1.3. If vector r ( t ) = f ( t ) vector i + g ( t ) vector j + h ( t ) vector k , then lim t a vector r ( t ) = lim t a f ( t ) vector i + lim t a g ( t ) vector j + lim t a h ( t ) vector k provided this limit exists....
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