Chapter 12.6, Problem 15PPSE

Solution

To transform the given vector and then writing the result in component form.

The component form of the vector is, $\langle 2,0\rangle .$

Given information:

  $\langle 0,2\rangle$ , rotate it ${270}^{\circ }.$

Concept:

First write the vector in matrix form.

Now for rotation, multiply the vector matrix with the given matrix corresponding to the degree of rotation.

${90}^{\circ }$	$\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$
${180}^{\circ }$	$\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$
${270}^{\circ }$	$\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$
${360}^{\circ }$	$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Calculation:

The vector in matrix form is:

  $\left[\begin{array}{c}0\\ 2\end{array}\right]$

Now multiply it with the matrix given in table above:

  $\begin{array}{c}\left[\begin{array}{c}0\\ 2\end{array}\right]\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]=\left[\begin{array}{c}0+2\\ 0+0\end{array}\right]\\ =\left[\begin{array}{c}2\\ 0\end{array}\right]\end{array}$

Hence the resultant vector in component form is $\langle 2,0\rangle .$

Conclusion:

Hence the resultant vector in component form is $\langle 2,0\rangle .$