For all unit vector $\nu \in \mathbb{R}^n$ consider an affine hyperplane $A_{\nu}$ orthogonal to the direction $\nu $.
Now consider n linearly independent unit vectors $\nu_ 1 , \nu_2, \dots, \nu_n \in \mathbb{R}^n$. I'm asking: is the intersection $A_{\nu_1} , A_{\nu_2}, \dots , A_{\nu_n}$ necessarily a point?
I think the answer is yes, but I'm not sure... In the case $n =2$ it is true (the intersection of two lines having different directions is a point). In the case $n=3$ is also true. But in general is it true?
Yes, it is just a point. It is a solution of a system$$\left\{\begin{array}{l}\langle\nu_1,x\rangle=\mu_1\\\langle\nu_2,x\rangle=\mu_2\\\vdots\\\langle\nu_n,x\rangle=\mu_n.\end{array}\right.$$This is a system of $n$ linear equations in $n$ unknowns and the fact that the $\nu_k$'s are linearly independent is equivalent to the assertion that the matrix of its coefficients is invertible. Therefore, the system has one and only one solution.