Exercise 8

A theme that will later unfold concerns the use of coordinate systems. We can identify the point \((x,y)\) with the tip of the vector \(\left[\begin{array}{r}x\\y\end{array}\right]\text{,}\) drawn emanating from the origin. We can then think of the usual Cartesian coordinate system in terms of linear combinations of the vectors

\begin{equation*} \evec_1 = \left[\begin{array}{r} 1 \\ 0 \end{array}\right], \evec_2 = \left[\begin{array}{r} 0 \\ 1 \end{array}\right]\text{.} \end{equation*}

Figure 2.1.8 The usual Cartesian coordinate system, defined by the vectors \(\evec_1\) and \(\evec_2\text{,}\) is shown on the left along with the representation of the point \((2,-3)\text{.}\) The right shows a nonstandard coordinate system defined by vectors \(\vvec_1\) and \(\vvec_2\text{.}\)

The point \((2,-3)\) is identified with the vector

\begin{equation*} \left[\begin{array}{r} 2 \\ -3 \end{array}\right] = 2\evec_1 - 3\evec_2\text{.} \end{equation*}

If we have vectors

\begin{equation*} \vvec_1 = \left[\begin{array}{r} 2 \\ 1 \end{array}\right], \vvec_2 = \left[\begin{array}{r} 1 \\ 2 \end{array}\right]\text{,} \end{equation*}

we may define a new coordinate system, such that a point \(\{x,y\}\) will correspond to the vector

\begin{equation*} x\vvec_1 + y\vvec_2\text{.} \end{equation*}

For instance, the point \(\{2,-3\}\) is shown on the right side of FigureĀ 2.1.8

  1. Write the point \(\{2,-3\}\) in standard coordinates; that is, find \(x\) and \(y\) such that

    \begin{equation*} (x,y) = \{2,-3\}\text{.} \end{equation*}
  2. Write the point \((2,-3)\) in the new coordinate system; that is, find \(a\) and \(b\) such that

    \begin{equation*} \{a,b\} = (2,-3)\text{.} \end{equation*}
  3. Convert a general point \(\{a,b\}\text{,}\) expressed in the new coordinate system, into standard Cartesian coordinates \((x,y)\text{.}\)

  4. What is the general strategy for converting a point from standard Cartesian coordinates \((x,y)\) to the new coordinates \(\{a,b\}\text{?}\) Actually implementing this strategy in general may take a bit of work so just describe the strategy. We will study this in more detail later.