Update main.tex

Author: annadavismath

VEC-0005/main.tex

@@ -28,7 +28,7 @@ \section*{Vectors and their Representations}
 
 \subsection*{How to Create a Coordinate System}
 
-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
+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 structure:
 \begin{itemize}
     \item Location of the origin
     \item Unit of length
@@ -73,7 +73,7 @@ \subsection*{How to Create a Coordinate System}
 \end{question}
 
 \begin{question}
-Move point $B$ to coincide with $P_3$
+REFRESH your browser to return to the original coordinate system. Move point $B$ to coincide with $P_3$
     List the coordinates for each $P_i$ with respect to the new coordinate system.  
     $$P_1=\left(\answer{1},\answer{-1}\right)$$
     $$P_2=\left(\answer{-2},\answer{0}\right)$$
@@ -110,11 +110,18 @@ \subsection*{How to Create a Coordinate System}
  \end{center}
 
  \begin{question}
-     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.
+     Suppose we want to express $P_1$ using a ``coordinate system" determined by three vectors $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\overrightarrow{OC}$.  Fill in the missing coordinates for $P_1$ below, if the coordinates are to be of the form $(\overrightarrow{OA}\text{-coordinate}, \overrightarrow{OB}\text{-coordinate}, \overrightarrow{OC}\text{-coordinate})$.
      $$P_1=\left(\answer{1},\answer{1},0\right)$$
      $$P_1=\left(0,\answer{2},\answer{0.5}\right)$$
      $$P_1=\left(\answer{2}, 0, \answer{-0.5}\right)$$
      $$P_1=\left(\answer{3},-1,-1\right)$$
+
+How many ways are there to express $P_1$ using this ``coordinate system"?
+\begin{multipleChoice}
+    \choice{1}
+    \choice{4}
+    \choice{Infinitely many}
+\end{multipleChoice}
  \end{question}
 
  % \begin{question}
@@ -127,7 +134,7 @@ \subsection*{How to Create a Coordinate System}
 
 \end{exploration}
 
-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.  
+What we discovered 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.  
 
 \begin{idea}
     What makes a good coordinate system?