Let \(Q=\left(\begin{array}{cc}\cos \frac{\pi}{4}-\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} & \cos \frac{\pi}{4}\end{array}\right)\) and \(x=\left(\begin{array}{c}\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{array}\right)\) then \(Q^{3} x\) is equal to

Question

Let \(Q=\left(\begin{array}{c}\cos \frac{\pi}{4}-\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} \cos \frac{\pi}{4}\end{array}\right)\) and \(x=\left(\begin{array}{c}\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{array}\right)\) then \(Q^{3} x\) is equal to
(A) \(\quad\left(\begin{array}{l}0 \\ 1\end{array}\right)\)
(B) \(\left(\begin{array}{c}-\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{array}\right)\)
(C) \(\left(\begin{array}{l}-1 \\ 0\end{array}\right)\)
(D) \(\left(\begin{array}{c}-\frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}}\end{array}\right)\)

Answer 1

Ans: (C)
If \(Q(\theta)=\left[\begin{array}{c}\cos \theta-\sin \theta \\ \sin \theta \\ \cos \theta\end{array}\right], Q^{3}(\theta)=Q(3 \theta), Q^{3}(\pi / 4)=\left[\begin{array}{cc}\cos 3 \pi / 4 & -\sin 3 \pi / 4 \\ \sin 3 \pi / 4 & \cos 3 \pi / 4\end{array}\right]=\left[\begin{array}{cc}-1 / \sqrt{2} & -11 / \sqrt{2} \\ 1 / \sqrt{2} & -1 / \sqrt{2}\end{array}\right], Q^{3}(\pi / 4) x=\left(\begin{array}{l}-1 \\ 0\end{array}\right)\)

Answer 2

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Answer 3

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