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What determines the positive/negative direction of a coordinate system? It is pre-defined that in the case of the XY plane for example, for $x\geq 0$, the X-axis is positive and for $y\geq 0$, the Y-axis is positive.

But if we define the angle between the axes to start from the positive Y-axis, we in practice rotated the axis by 90 degrees and now for $x\leq 0$, is the X-axis positive?

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  • $\begingroup$ It's basically a matter of convention. So, you can choose which direction is positive, but you usually shouldn't choose the uncommon one because it would be confusing. $\endgroup$ – shardulc Apr 27 '16 at 9:28
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Actually, it's not pre defined that the right hand side of the X axis is positive and the upward Y direction is positive.

For example, a lot of the time in computer science, we take the bottom Y direction as positive (since text flows from top to bottom, it makes it easier to think of it that way).

However, if you arbitrarily choose to pick the +ve x-axis to be the right direction and the +ve y-axis to be upward, then you can take the matrix that corresponds to your rotation by 90 degrees, make it act on the basis vectors $(1, 0)$ and $(0, 1)$, then seee what that gets us.

Since the 2-d rotation matrix $R(\theta)$ that rotates the plane by $\theta$ degrees clockwise is given by

$$R(\theta) = \begin{bmatrix} cos(\theta) &sin(\theta)\ \\ -sin(\theta) & cos(\theta)\end{bmatrix}$$

substitute $\theta = 90^\circ$ to give $$ T = R(90^{\circ}) = \begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix}$$

We can see that $T (1, 0) = (0, 1) \\ T(0, 1) = (-1, 0)$

and hence, the +x axis $(1, 0)$ goes to the +y axis $(0, 1)$ and the +y axis $(0, 1)$ goes to the -x axis $(-1, 0)$

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