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What are some typical applications that lead to n-tuples?
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Here are some typical applications that lead to n-tuples.

• Experimental Data - A scientist performs an experiment and makes $n$ numerical measurements each time the experiment is performed. The result of each experiment can be regarded as a vector $\mathbf{y}=\left(y_{1}, y_{2}, \ldots, y_{n}\right)$ in $R^{n}$ in which $y_{1}, y_{2}, \ldots, y_{n}$ are the measured values.
• Storage and Warehousing - A national trucking company has 15 depots for storing and servicing its trucks. At each point in time the distribution of trucks in the service depots can be described by a 15-tuple $\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{15}\right)$ in which $x_{1}$ is the number of trucks in the first depot, $x_{2}$ is the number in the second depot, and so forth.
• Electrical Circuits - A certain kind of processing chip is designed to receive four input voltages and produce three output voltages in response. The input voltages can bé régarded as vectors in $R^{4}$ and thẻ output voltagess *** vectors in $R^{3}$. Thus, thè chip can be viewed as a device that transforms an input vector $\mathbf{v}=\left(v_{1}, v_{2}, v_{3}, v_{4}\right)$ in $R^{4}$ into an output vector $\mathbf{w}=\left(w_{1}, w_{2}, w_{3}\right)$ in $R^{3}$.
• Graphical Images - One way in which color images are created on computer screens is by assigning each pixel (an addressable point on the screen) three numbers that describe the hue, saturation, and brightness of the pixel. Thus, a complete color image can be viewed as a set of 5 -tuples of the form $\mathbf{v}=(x, y, h, s, b)$ in which $x$ and $y$ are the screen coordinates of a pixel and $h, s$, and $b$ are its hue, saturation, and brightness.
• Economics - One approach to economic analysis is to divide an economy into sectors (manufacturing, services, utilities, and so forth) and measure the output of each sector by a dollar value. Thus, in an economy with 10 sectors the economic output of the entire economy can be represented by a 10-tuple $\mathbf{s}=\left(s_{1}, s_{2}, \ldots, s_{10}\right)$ in which the numbers $s_{1}, s_{2}, \ldots, s_{10}$ are the outputs of the individual sectors.
• Mechanical Systems - Suppose that six particles move along the same coordinate line so that at time $t$ their coordinates are $x_{1}, x_{2}, \ldots, x_{6}$ and their velocities are $v_{1}, v_{2}, \ldots, v_{6}$, respectively. This information can be represented by the vector
$$\mathbf{v}=\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}, t\right)$$
in $R^{13} .$ This vector is called the state of the particle system at time $t$.
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