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The rotation transofrmation is defined as some composition of rotatation along the $x,y,z$ axes.

Assuming $T$ is a rotation transformation in $\mathbb{R}^{3}, v=\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right), T\left(v\right)=\left(1,0,0\right)$

I need to find the matrix of $T$ according to the standard base. I was trying to find a rotation through $y$ axis such as $S_{\phi}\left(v\right)\subseteq XY$ plane, where $S$ is rotation along $y$-axis. Not sure if they meant $T$ is rotation along $z$-axis. But I am not sure if that's how I handle this question.

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  • $\begingroup$ There seems to be insufficient information; also the vectors you have given us seems to be transposes of what you intended. $\endgroup$
    – user21436
    Jan 15, 2012 at 13:05
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    $\begingroup$ @KannappanSampath: I find your second remark a bit pedantic: certainly there is nothing wrong with denoting an element of $\mathbb R^3$ by a triplet of real numbers. The habit of writing them vertically helps us remember how to operate by a matrix on a vector, but I would not advocate imposing the use of distinct versions of $\mathbb R^3$ to contain column vectors and row vectors. $\endgroup$
    – Marc van Leeuwen
    Jan 15, 2012 at 13:45

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Your equations are insufficient to determine the rotation $T$. To see this, consider a rotation $R$ with axis $\left<v\right>$; if $T$ is any solution then so is $T\circ R$, since compositions of rotations are again rotations. (You can also compose on the right with rotations with axis $\left<T(v)\right>$, although this gives no other alternative solutions than those found by the first method.)

The axis of any rotation that solves this problem must lie in the reflection plane (perpendicular bisector) $H$ between the vectors $v$ and $T(v)$, since any point of the axis obviously stays at equal distances form the two. Moreover any line $l$ in $H$ and passing through the origin can be the axis of a rotation solving the problem. To see this, compose the reflection in $H$ with the unique reflection fixing both $l$ and $T(v)$ (i.e., reflection in the plane spanned by $l$ and $T(v)$). This composition is a rotation, and it sends $v$ to $T(v)$.

For instance, one gets a rotation by a minimal angle by choosing $l$ to be perpendicular to both $v$ and $T(v)$: the line spanned by $(0,1,-1)$. Compute the reflection in $H$ followed by the one in the plane spanned by $(1,0,0)$ and $(0,1,-1)$ (the latter is simple: $(x,y,z)\mapsto(x,-z,y)$) to find a concrete solution.

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