Equation of an $(n-2)-$sphere in $\mathbb{R}^n$.

Question

I am looking for the equation of an $(n-2)$-sphere in $\mathbb{R}^{n}$ generated from the intersection of the $(n-1)$- sphere $x_1^2 + x_2^2 + \cdots + x_n^2 = r^2$, and the hyperplane perpendicular to a given vector $ \mathbf{v} \in \mathbb{R}^{n}$ and containing the point $(0,0,\ldots,0)$.

I just need some hint to solve this problem. Thanks for any help!

Answer 1

The additional condition is $\mathbf{v}\cdot (x_1,\dots,x_n) = 0$.