Direction of vector curve How does one generally determine the direction of a vector-valued curve? For example, does $$\vec{g}(t) = (t \sin(t), t \cos(t), \sqrt3 t)$$ point counterclockwise? What makes me ask this question is the order of the trig. parameters for $x$ and $y$. I know that if they were interchanged then the curve would definitely point counterclockwise. I'm a little confused.
 A: "Clockwise or anticlockwise" has meaning only when the plane of the clock-face  being referred to is known.In the case of cone you gave ( cone semi vertical angle $ \pi/6 $) if you specify the plane in which $t$ is measured with $x,y$ it makes sense. By convention anticlockwise. If z-axis of cone is brought in, we have to see which plane of reference $ XY,YZ,ZX  $ we are referencing to determine direction of vector.
A: One can ask about whether the parameterized curve ${\bf g}(t)$ appears to be moving clockwise or counterclockwise, but note that (for curves in $3$-space) this depends on one's literal perspective: (For $t > 0$) this particular curve appears to be moving clockwise when looking down the positive $z$-axis but anticlockwise when look up the negative $z$-axis.
We can make "appear to be moving [e.g.] clockwise" a little more precise by defining the following:


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*A $(C^2)$ parameterized curve ${\bf h}(t)$ in $\Bbb R^2$ is anticlockwise at a time $t_0$ if the signed curvature $\kappa(t_0)$ of ${\bf h}(t)$ at time $t_0$ is positive and clockwise if it is negative.

*A parameterized curve ${\bf g}(t) = (x(t), y(t), z(t))$ in $\Bbb R^3$ appears to be moving anticlockwise at a time $t_0$ (from the positive $z$-axis) if the signed curvature of the curve $(x(t), y(t))$ (which we can identify with the projection of ${\bf g}(t)$ to the $xy$-plane) is anticlockwise at $t_0$ and ...clockwise... if it is negative.


When computing this for explicit curves, it's convenient to know that the sign of the signed curvature is the same as the sign of $$\det \pmatrix{{\bf h}'(t) & {\bf h}''(t)} .$$ In particular, this shows that the property of being anticlockwise/clockwise does not change under orientation-preserving reparameterization but reverses under orientation-reversing reparameterization, and so is a property of an oriented curve (and does not depend on a particular oriented parameterization). (Here, we of course take all reparameterizations to be $C^2$.)
There is a notion for ($C^3$) curves in $\Bbb R^3$ analogous to signed curvature that does not depend on a choice of perspective (like "...from the positive $z$-axis"), namely the sign of the torsion $\tau(t)$ at a given time $t_0$. Again, it's convenient to know that this sign agrees with the sign of $$\det \pmatrix{{\bf g}'(t) & {\bf g}''(t) & {\bf g}'''(t)} .$$ Unlike signed curvature, this property is independent under all ($C^3$) reparameterizations, both orientation-preserving and not, and so is a property of the curve itself, and does not depend on a choice of parameterization or even orientation. I don't know that there is a standard terminology for this notion beyond positive/negative torsion, though something like right-handed/left-handed twisting would be appropriate.
