Need the parametric equation of a circle perpendicular to a vector.

[okkvlt]

i need a parametric equation of a circle in 3d space that is perpendicular to a vector <a,b,c>. (as t goes up the circle is traced counterclockwise, as viewed from the head of the vector.)
in the form x[t],y[t],z[t]
i know that x^2+y^2+z^2=constant
and that ax+by+cz=0

But i cannot figure out the parametric equation x[t],y[t],z[t] that describes a circle perpendicular to the vector.

 

[okkvlt]

or, phrased in other words, this is the intersection of the plane ax+by+cz=0 and the sphere x^2+y^2+z^2=constant.


in case anybodys wondering, I am working on stokes theorem.

 

[LCKurtz]

There may be a shorter way in some specific cases, but you might try this. Solve the plane for z and put that in the equation of the sphere. This will give you an xy equation which represents the projection of the intersection curve in the xy plane. This will be an ellipse. Complete the square on it and get it in the standard form:

$$\frac {(x-p)^2}{a^2} + \frac {(y-q)^2}{b^2} = 1$$

Then you can parameterize it as:

$$x = p + a\cos(t)\ y=q + b\sin(t)$$

and use these to get z on the plane in terms of t also.

 

[LCKurtz]

I want to add, after thinking about my reply, that it isn't so simple. The equation in the xy plane will likely be both translated and rotated. And since the OP is working with Stoke's Theorem, my guess is that the circuit integral might be unnecessary and, depending on the specific problem, the surface integral that Stoke's theorem gives may be easy. Hard to say without seeing the specific problem.

 

[okkvlt]

i want to prove that the magnitude of curl is the line integral around a region perpendicular to the curl vector.