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?

  • $\begingroup$ 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

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 $(-1, 0)$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.