Let $u_1, u_2, v\in \mathbb{R}^n$ be linearly independent vectors but not orthogonal.
Set $w_1= \langle u_1, v \rangle \hat{i}$ and $w_2 = \langle u_2, v \rangle\hat{j}$ where $\hat{i}$ and $\hat{j}$ are defined in the usual way as orthogonal unit vectors in $\mathbb{R}^2$.
In addition, suppose we take $w_1$ and $w_2$ and transform them in the following way $x_1=\frac{w_1}{\sqrt{\|w_1\|^2+ \|w_2\|^2}}$ and $x_2=\frac{w_2}{\sqrt{\|w_1\|^2+ \|w_2\|^2}}$ so that $x_1+x_2$ lies on the unit circle.
- Is there a geometric interpretation of $x_1$ and $x_2$ in terms of the original vectors $u_1$, $u_2$ and $v$?
For instance, I know $\operatorname{span}(u_1,u_2)$ should define a plane. Additionally, $v$ would form a pyramid with $u_1$ and $u_2$. The inner products between $v$ and $u_1$ and $v$ and $u_2$ are related to the angle between each pair of vectors.
Additionally, if we form a triangle using $x_1$ and $x_2$ where $\theta$ is the angle opposite $x_1$ and adjacent to $x_2$ then $\tan\theta$ is the ratio of $\langle u_1, v \rangle$ to $\langle u_2, v \rangle$ which is proportional to the ratio of
$$\|\operatorname{proj}_{u_1}(v)\|$$
to
$$\|\operatorname{proj}_{u_2}(v)\|$$
Any additional geometric insights would be greatly appreciated.
- For fixed $u_1$ and $u_2$, if we know the value of $\left< u_1, v \right>$, are there any constraints on the value of $\left< u_2, v \right>$? If so, what are they?
$ \newcommand\form[1]{\langle#1\rangle} $I don't think you're going to get anything out of question 1, since injecting $\hat i, \hat j$ into the problem is very artificial.
As for question 2, $$ (u_1)^\perp_a = \{v \;:\; \form{u_1, v} = a\} $$ is a hyperplane (the hyperplane $u_1$ orthogonal to $u_1$ and a distance $a$ from the origin). Then $(u_2)^\perp_b$ must intersect $(u_1)^\perp_a$ non-trivially for any $b$ since $u_1, u_2$ are linearly independent; so $b$ can be any value. More concretely, just consider the 2D case; for what values of $b$ does the system $$ v^xu^x_1 + v^yu^y_1 = a,\quad v^xu^x_2 + v^yu^y_2 = b $$ have a solution for $v^x, v^y$? For all $b$, since the columns of $$ \begin{pmatrix} u_1^x & u_1^y \\ u_2^x & u_2^y \end{pmatrix} $$ are linearly independent, and so the matrix is invertible.