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‎VEC-0005/main.tex‎

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\documentclass{ximera}
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\input{../preamble.tex}
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\title{Vectors and their Representations} \license{CC BY-NC-SA 4.0}
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\begin{document}
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\begin{abstract}
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\end{abstract}
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\maketitle
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\begin{onlineOnly}
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\section*{Vectors and their Representations}
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\end{onlineOnly}
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We will start our Linear Algebra journey with a very important point.
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\begin{center}
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\begin{tikzpicture}
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\fill (0,0) circle (2pt) node[above right] {$P=$ very important point};
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\end{tikzpicture}
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\end{center}
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How would you describe the location of $P$? Whether you reference the edges of the page or introduce your own set of axes, you will find that (1) a fixed reference system is required to describe the location, and (2) the resulting description may differ from someone else’s, since different choices of coordinate system lead to different representations.
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\begin{idea}
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A mathematical object is distinct from its representation. A single object may have a multitude of representations that depend on convention and convenience.
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\end{idea}
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\subsection*{How to Create a Coordinate System}
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When you first encountered vectors in your earlier courses it was probably assumed that these vectors exist in some established rectangular coordinate system. Such a coordinate system implicitly postulates the following
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\begin{itemize}
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\item Location of the origin
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\item Unit of length
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\item Orthogonality of axes
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\end{itemize}
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Representation of points and vectors was dictated by the established coordinate system.
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What if we started from scratch? Take two vectors, choose the location of the origin, then use the two vectors to create a coordinate grid. The interactive below allows you to move points $A$ and $B$ to create two vectors. Move the slider to see how different points in the plane can be represented using this coordinate grid, and how the representation of a fixed point changes depending on what grid is chosen.
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% https://www.geogebra.org/classic/edcyushm
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\begin{center}
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\geogebra{edcyushm}{800}{600}
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\end{center}
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When using an alternative coordinate system, it is customary to identify the vectors that determine it. For now, we will say that the coordinates of point $P$ are stated with respect to $\{\overrightarrow{OA}, \overrightarrow{OB}\}$. We will refine this statement later in the text.
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\begin{exploration}\label{exp:coordSystemLinCombs1}
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Use the interactive below to answer questions about the coordinates of points with respect to various coordinate systems.
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% https://www.geogebra.org/classic/qw5dpmqq
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\begin{center}
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\geogebra{qw5dpmqq}{800}{600}
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\end{center}
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\begin{question}
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List the coordinates for each $P_i$ with respect to the given coordinate system. Your coordinates should be of the form $(\overrightarrow{OA}\text{-coordinate}, \overrightarrow{OB}\text{-coordinate})$.
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$$P_1=\left(\answer{1},\answer{1}\right)$$
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$$P_2=\left(\answer{-2},\answer{0}\right)$$
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$$P_3=\left(\answer{0},\answer{-1}\right)$$
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$$P_4=\left(\answer{-2},\answer{3}\right)$$
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$$P_5=\left(\answer{3},\answer{-2}\right)$$
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\end{question}
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\begin{question}
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REFRESH your browser to return to the original coordinate system. Move point $A$ to coincide with $P_1$
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List the coordinates for each $P_i$ with respect to the new coordinate system.
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$$P_1=\left(\answer{1},\answer{0}\right)$$
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$$P_2=\left(\answer{-2},\answer{2}\right)$$
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$$P_3=\left(\answer{0},\answer{-1}\right)$$
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$$P_4=\left(\answer{-2},\answer{5}\right)$$
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$$P_5=\left(\answer{3},\answer{-5}\right)$$
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How do these coordinates compare to the coordinates in the original coordinate system? Explain why this is happening. (Hint: REFRESH your browser to return to the original coordinate system to compare.)
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\end{question}
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\begin{question}
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Move point $B$ to coincide with $P_3$
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List the coordinates for each $P_i$ with respect to the new coordinate system.
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$$P_1=\left(\answer{1},\answer{-1}\right)$$
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$$P_2=\left(\answer{-2},\answer{0}\right)$$
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$$P_3=\left(\answer{0},\answer{1}\right)$$
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$$P_4=\left(\answer{-2},\answer{-3}\right)$$
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$$P_5=\left(\answer{3},\answer{2}\right)$$
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How do these coordinates compare to the coordinates in the previous question? Explain why this is happening. (Hint: REFRESH your browser to return to the original coordinate system to compare.)
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\end{question}
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% \begin{question}
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% REFRESH your browser to return to the original coordinate system. Express each $\overrightarrow{OP}_i$ as a linear combination of $\overrightarrow{OA}$ and $\overrightarrow{OB}$.
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% $$\overrightarrow{OP}_1=\answer{1}\overrightarrow{OA}+\answer{1}\overrightarrow{OB}$$
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% $$\overrightarrow{OP}_2=\answer{-2}\overrightarrow{OA}+\answer{0}\overrightarrow{OB}$$
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% $$\overrightarrow{OP}_3=\answer{0}\overrightarrow{OA}+\answer{-1}\overrightarrow{OB}$$
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% $$\overrightarrow{OP}_4=\answer{-2}\overrightarrow{OA}+\answer{3}\overrightarrow{OB}$$
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% $$\overrightarrow{OP}_5=\answer{3}\overrightarrow{OA}+\answer{-2}\overrightarrow{OB}$$
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% Discuss the relationship between your answers to the first question and your answers here.
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% \end{question}
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\begin{question}
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Move point $B$ to coincide with $P_2$. What do you observe? Do vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ determine a good coordinate system for the plane? Can we express every point in the plane with respect to the coordinate system determined by $\{\overrightarrow{OA}, \overrightarrow{OB}\}$? Can we express \textit{some} points in the plane with respect to the coordinate system determined by $\{\overrightarrow{OA}, \overrightarrow{OB}\}$?
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\end{question}
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\end{exploration}
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\begin{exploration}\label{exp:coordSystemLinCombs2}
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We will use the same set-up as in the previous exploration but introduce an additional vector $\overrightarrow{OC}$.
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% https://www.geogebra.org/classic/b3k96x2w
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\begin{center}
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\geogebra{b3k96x2w}{800}{600}
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\end{center}
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\begin{question}
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Suppose we want to express $P_1$ using a coordinate system determined by three vectors $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\overrightarrow{OC}$. If the coordinates are to be of the form $(\overrightarrow{OA}\text{-coordinate}, \overrightarrow{OB}\text{-coordinate}, \overrightarrow{OC}\text{-coordinate})$, how many ways do you think there would be to express $P_1$? Fill in the missing coordinates for $P_1$ below.
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$$P_1=\left(\answer{1},\answer{1},0\right)$$
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$$P_1=\left(0,\answer{2},\answer{0.5}\right)$$
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$$P_1=\left(\answer{2}, 0, \answer{-0.5}\right)$$
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$$P_1=\left(\answer{3},-1,-1\right)$$
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\end{question}
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% \begin{question}
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% Based on your work above, express $\overrightarrow{OP_1}$ as a linear combination of $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\overrightarrow{OC}$. How many ways do you think there are to do this?
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% \end{question}
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\begin{question}
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Compare and contrast the coordinate systems in this exploration and Exploration \ref{exp:coordSystemLinCombs1}.
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\end{question}
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\end{exploration}
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What we learned is that not all collections of vectors are adequate for making a useful coordinate system. In Exploration \ref{exp:coordSystemLinCombs1} we learned that if two vectors are collinear, then not all points in the plane can be represented with respect to those two vectors. In Exploration \ref{exp:coordSystemLinCombs2} we found that having too many vectors results in a point having infinitely many representations, which would be computationally confusing.
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\begin{idea}
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What makes a good coordinate system?
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\begin{itemize}
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\item Every point has coordinates associated with it;
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\item The set of coordinates associated with each point is unique.
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\end{itemize}
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\end{idea}
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\end{document}

‎sandbox.tex‎

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\activity{SAND-0010/main}
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\activity{VEC-0005/main}
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\end{document}

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