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# help

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The point $$(r,\theta)$$ in polar coordinates is $$(7,5)$$ in rectangular coordinates. What is the point $$\left( 2r, \theta + \frac{\pi}{2} \right)$$ in rectangular coordinates?

Nov 15, 2019

#1
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The point $$(r,\theta)$$ in polar coordinates is $$(7,5)$$ in rectangular coordinates.
What is the point $$\left( 2r, \theta + \dfrac{\pi}{2} \right)$$  in rectangular coordinates?

$$\begin{array}{|rcll|} \hline && \text{Set (7,5) to 7+5i } \\\\ && \text{Rotation by } \dfrac{\pi}{2}: \qquad \times i \\ && \text{scaling} : \qquad \times 2 \\\\ && 2\times(7+5i)\times i \\ &=& 2\times(7i+5i^2) \quad | \quad i^2 = -1 \\ &=& 2\times(7i-5) \\ &=& -10+14i \\\\ && \mathbf{\left( 2r, \theta + \dfrac{\pi}{2} \right) = (-10,14)} \\ \hline \end{array}$$

Nov 15, 2019