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When performing rotations in three space, is a rotation from x to z considered positive or negative? How do you determine whether similar rotations in three space are positive or negative?

okarin
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It can be positive or negative depends on the "direction" of your rotation axis. The convention is the right hand rule.

If you point your thumb in the "direction" of your rotation axis and you index finger to the direction of the $1^{st}$ vector. If you can point your middle finger to the direction of the $2^{nd}$ vector, then the rotation is considered to be positive. If not, then the rotation is considered to be negative.

In ordinary $\mathbb{R}^3$, the "positive directions" of the 3 axis is setup such that if you rotate the vector in "positive direction" of $x$-axis with respect to the "positive direction" of $z$-axis for $90^{\circ}$, you get a vector in the "positive direction" of $y$-axis.

If you attempt to rotate the vector in "positive direction" of $x$-axis with respect to the "positive direction" of $y$-axis for $90^{\circ}$, you get a vector in the "negative direction" of $z$-axis.

So the answer to your original question is "negative" when you measure your rotation with respect to the "positive direction" of $y$-axis.

Sound confusing and hard to remember? Same to me. When in doubt, I always use my right hand to figure out the "sign" of a rotation.

achille hui
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