Question 10.4.3: Relating the coordinates Suppose that the x- and y-axes are ...
Relating the coordinates
Suppose that the x– and y-axes are rotated through the angle π/3. If the point P has coordinates (2, 5) in the xy-system, then what are the coordinates of P in the x’y’-system?
Use the rotation equations x^{\prime}=x \cos \theta+y \sin \theta and y^{\prime}=-x \sin \theta+y \cos \theta with \theta=\pi / 3 and (x, y) = (2, 5):
\begin{aligned} &x^{\prime}=2 \cos \frac{\pi}{3}+5 \sin \frac{\pi}{3}=2 \cdot \frac{1}{2}+5 \cdot \frac{\sqrt{3}}{2}=\frac{2+5 \sqrt{3}}{2} \\ &y^{\prime}=-2 \sin \frac{\pi}{3}+5 \cos \frac{\pi}{3}=-2 \cdot \frac{\sqrt{3}}{2}+5 \cdot \frac{1}{2}=\frac{5-2 \sqrt{3}}{2} \end{aligned}
So in x^{\prime} y^{\prime}-system P has coordinates \left(\frac{2+5 \sqrt{3}}{2}, \frac{5-2 \sqrt{3}}{2}\right).
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