I think this is a commonly asked question but I haven't found a satisfying answer to this.

Say we have unit non-negative vectors $w_1,w_2 \in \mathbb{R}^2$. I want to get all the vectors $x$ on the unit circle between $w_1$ and $w_2$. One way to trace the path from $w_1,w_2$ would be to use Givens rotation matrices to "rotate" $x$ from $w_1$ to $w_2$. As an example, say $w_1$ subtends angle $\theta_1$ with the first axis and $w_2$ subtends $\theta_2 > \theta_1$. Then I can rotate $x$ from $w_1$ to $w_2$ by using the following rotation for all angles $\alpha \in \left[0,\theta_2-\theta_1\right]$.

$$ \begin{bmatrix} \cos{\alpha} &-\sin{\alpha} \\ \sin{\alpha} & \cos{\alpha} \end{bmatrix} \begin{bmatrix} \cos{\theta_1} \\ \sin{\theta_1} \end{bmatrix} = \begin{bmatrix} \cos{(\theta_1 + \alpha)} \\ \sin{(\theta_1 + \alpha)} \end{bmatrix} $$

Can we generalise this to $\mathbb{R}^n$. Say we have unit non-negative vectors $w_1, w_2, \ldots ,w_n \in \mathbb{R}^n$. These vectors would form a surface on the unit ball. Is there a parameterised way I can "rotate" a vector to trace out this surface. What do these rotation matrices look like?


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