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December 13th, 2016, 07:59 PM   #1
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Unhappy Plane Coordinates

The attached figure illustrates a rectangular coordinate system xy generated by the unit vectors of the canonical base $\displaystyle \vec{i}$ and $\displaystyle \vec{j}$ and a system of coordinates non-rectangular $\displaystyle {x}'{y}'$ generated by the unit vectors $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$ ($\displaystyle \left \| \vec{u} \right \| = \left \| \vec{v} \right \| = 1$), where the axis $\displaystyle {y}'$ coincides with the y-axis, while $\displaystyle {x}'$ is obtained from its anti-clockwise rotation by an angle $\displaystyle \theta = \frac{\pi}{6}$. Find the $\displaystyle {x}'{y}'$ coordinates of the points whose coordinates $\displaystyle xy$ are $\displaystyle (10,-3)$. Use $\displaystyle cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$ and $\displaystyle sin(\frac{\pi}{6}) = \frac{1}{2}$.( Remember that $\displaystyle (x,y) = x\vec{i}+y\vec{j} = {x}'\vec{u}+{y}'\vec{v}$)
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December 13th, 2016, 09:15 PM   #2
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The attached figure illustrates a rectangular coordinate system xy generated by the unit vectors of the canonical base $\displaystyle \vec{i}$ and $\displaystyle \vec{j}$ and a system of coordinates non-rectangular $\displaystyle {x}'{y}'$ generated by the unit vectors $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$ ($\displaystyle \left \| \vec{u} \right \| = \left \| \vec{v} \right \| = 1$), where the axis $\displaystyle {y}'$ coincides with the y-axis, while $\displaystyle {x}'$ is obtained from its anti-clockwise rotation by an angle $\displaystyle \theta = \frac{\pi}{6}$. Find the $\displaystyle {x}'{y}'$ coordinates of the points whose coordinates $\displaystyle xy$ are $\displaystyle (10,-3)$. Use $\displaystyle cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$ and $\displaystyle sin(\frac{\pi}{6}) = \frac{1}{2}$.( Remember that $\displaystyle (x,y) = x\vec{i}+y\vec{j} = {x}'\vec{u}+{y}'\vec{v}$)
What have you been able to do so far?

-Dan
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December 14th, 2016, 03:01 AM   #3
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What have you been able to do so far?

-Dan
I have not done anything yet, I do not know where to start.
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December 14th, 2016, 05:57 AM   #4
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Seeing that this involves angles and you get coordinates of a points in a given coordinate system by drawing perpendiculars from the point to the coordinate axes, I would start by drawing those lines and using trigonometry. (Do you suppose that's why they gave you the sine and cosine of $\displaystyle \pi/6$?)
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