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How do I specify the directions of a unit vector $X_n$ and $Y_n$ of a frame $\{n\}$ at an angle $θ$ relative to a frame $\{a\}$ in the form $$ X_n=cosθX_a + sinθY_a $$ $$ Y_n=-sinθX_a+cosθY_a $$ ? Can someone please help how the above equatione were derived?
Here's a graphical answer. Given some vector $r$ in Frame 1:
You can rotate to a new frame, F2:
All you do is project to the new axes. Counting along the x-axis in $F_2$, you can see that you go right (positive) in an amount $x_a \cos{\theta}$, then you go right (positive) in the amount $y_a \sin{\theta}$, so the total x-axis location is
$$ x_{F_2} = +\left(x_a\cos{\theta}\right) + \left(y_a\sin{\theta}\right) $$
Similarly, count "down" (negative) an amount $x_a \sin{\theta}$ and then count up (positive) an amount $y_a \cos{\theta}$ and get:
$$ y_{F_2} = -\left(x_a\sin{\theta}\right) + \left(y_a\cos{\theta}\right) $$
