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Suppose I have two unit vectors $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$ in 3D space that are perpendicular to each other. (Their dot product is zero.)

I want to rotate the $x$ vector by $\theta$ degrees such that it remains perpendicular to $y$. That is, I want to rotate $x$ using $y$ as the axis.

What would be the formula to do this and how is it derived?

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I will give you a general formula, where $\vec x$ and $\vec y$ are not necessarily unit vectors.

Assume $\Vert\vec x\Vert, \Vert\vec y\Vert \not= 0$ and they are not collinear. Call the resulting (rotated) vector $\vec x'$. First, let's get a vector that is perpendicular to $\vec y$. One such vector is $\vec x\times \vec y$, but for convenience, we will consider the vector $\vec z = \frac{\vec x\times \vec y}{\Vert\vec y\Vert}$. Note that $\Vert \vec z \Vert=\Vert \vec x \Vert$.

This vector is perpendicular to $\vec x$ as well. Because $\vec x'$ lies in the same plane as $\vec x$ and $\vec z$, it can be expressed as their linear combination, i.e.

$$\vec x' = \alpha \vec x + \beta \vec z$$ Multiplying (dot product) both sides with $\vec x$, we get

$$\vec x \cdot \vec x' = \alpha \vec x^2 + \beta \vec z \cdot \vec x = \alpha \vec x^2 \tag1$$ We used the fact that $\vec z$ is perpendicular to $\vec x$, making their dot product zero. Because $\Vert \vec x' \Vert = \Vert \vec x \Vert$, we have that $\vec x \cdot \vec x' = \Vert \vec x \Vert^2\cos\theta$. This, combined with $(1)$ gives us $\alpha = \cos\theta$. Using the fact that $\Vert \vec z \Vert = \Vert \vec x \Vert$ it can easily be shown that $\beta=\sin\theta$.

So, your vector is $$\vec x' = \cos\theta\cdot \vec x + \sin\theta\cdot \vec z$$

Notice that I haven't mentioned in which direction the vector is rotated. I will leave this for you to figure out.

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