Constrained optimization involving unit vectors

Question

I want to find the vector $v$ with the following properties

$v \cdot v = 1$, (unit length)

$v \cdot w = \alpha$, where $w$ is also a unit vector

$(v \cdot n)^2$ is minimized

Is there a closed solution for $v$?

Answer 1

WLOG assume $w=(1,0,...,0)$ and $n = (n_1,n_2,0,...,0)$; i.e. pick coordinates so that $w$ is the first unit vector, and $n$ is in the plane spanned by the first and second unit vectors. Write $v=(v_1,...,v_k)$.

Then $v\cdot w=\alpha$ means $v = (\alpha,v_2,...,v_k)$ and $v\cdot v=1$ means that $v_2^2+\cdots + v_k^2 = 1-\alpha^2$. Thus, if $\alpha > 1$ there is no solution.

Now note that, $$ (v\cdot n)^2 = (\alpha n_1 + v_2n_2)^2 = (\alpha n_1 / n_2 + v_2)^2 / n_2^2 $$

This is a quadratic in $v_2$ which is minimized at $v_2=\alpha n_1 / n_2$ and increases away from this value. However, we have the constraint $v_2^2 = 1 - \alpha^2 - v_3^2 - \cdots v_k^2$ which means we need $v_2^2 \leq 1 - \alpha^2$.

Thus, we should pick $v_2$ as close to $\alpha n_1 / n_2$ as possible while still keeping $v_2^2 \leq 1-\alpha^2$.

Explicitly, $$ v_2 = \begin{cases} -\sqrt{1-\alpha^2} & \alpha n_1/n_2 \leq -\sqrt{1-\alpha^2} \\ \alpha n_1 / n_2 & -\sqrt{1-\alpha^2} < \alpha n_1 / n_2 < \sqrt{1-\alpha^2} \\ \sqrt{1-\alpha^2} & \alpha n_1/n_2 \geq \sqrt{1-\alpha^2} \end{cases} $$

In the first and last cases, $v_3,...,v_k$ must all be zeero to satisfy the constraints. However, in the middle case we can pick any $v_3,...,v_k$ so that the constraints are satisfied (for $k>3$ there will be infinitely many solutions).