# Reference frame, vector

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:
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:
$$y_{F_2} = -\left(x_a\sin{\theta}\right) + \left(y_a\cos{\theta}\right)$$