Continuation of Example 10 The xy-coordinate system is rotated through the angle to
Chapter 7, Problem 7.1.1.161(choose chapter or problem)
Continuation of Example 10 The xy-coordinate system is rotated through the angle to obtain the \(x^{\prime} y^{\prime}\) -coordinate system (see Figure 7.11).
(a) Show that the inverse of the matrix
(b) Prove that the \((x, y)\) coordinates of P in Figure 7.11 are related to the \(\left(x^{\prime}, y^{\prime}\right)\) coordinates of P by the equations
\(\begin{array}{l} x=x^{\prime} \cos \alpha-y^{\prime} \sin \alpha \\ y=x^{\prime} \sin \alpha+y^{\prime} \cos \alpha \end{array}\).
(c) Prove that the coordinates \((x, y)\) can be obtained from the \(\left(x^{\prime}, y^{\prime}\right)\) coordinates by matrix multiplication. How is this matrix related to A?
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