Parametric equation of a circle that is normal to a vector, in nth dimension

Question

Given, a vector $A=(a,b,c,d,\dots)$ [ $n$ components] I am trying to come up with a general parametric equation of a circle such that, the given vector is normal to the plane containing the circle.

Any hint or suggestion as to how I can go about formulating this would be much appreciated.

Thank you.

Answer 1

Since you're in $n$ dimensions, you're not going to have a circle but an $(n-1)$-dimensional hypersphere:

Given an $n$-dimensional vector $A$ I am trying to come up with a general parametric equation of a hypersphere such that, the given vector is normal to the hyperplane containing the hypersphere.

Find the hyperplane $A^\perp$ normal to $A$, and let $\{u_1, u_2, \ldots, u_{n-1}\}$ be a basis for $A^\perp$. Now consider the equation $$\sum_{i=1}^{n-1}u_i^2=1.$$

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To restrict your shape to a ($2$-dimensional) circle, let all but two of the $u_i$ be zero. For example, $\{u_1^2 + u_2^2 = 1, u_3=u_4=\ldots=u_{n-1}=0\}$. Bear in mind that there will be infinitely many ways to find the $u_i$ in the first place, so this is a method to find a circle, not all such circles.