<s>[INST] Parametrise the circle centered at $ \ (1,1,-1) \ $ with radius equal to $ 3 $ Parametrise the circle centered at $ \ (1,1,-1) \ $ with radius equal to $ 3  $ in the plane $ x+y+z=1 $ with positive orientation . $$ $$ I have thought the parametriation: 
\begin{align} x(t)=1+ 3 \cos (t) \hat j +3 \sin (t)  \hat k \\ y(t)=1+3 \cos (t) \hat i+3 \sin (t) \hat k \\ z(t)=-1+3 \cos (t) \hat i +3 \sin (t) \hat j  , \ \ 0 \leq t \leq 2 \pi \end{align}  But I am not sure . Any help is there ?
 [/INST] $\mathbf u = (\frac {\sqrt 2}{2} \mathbf i - \frac {\sqrt 2}{2} \mathbf j)\\
\mathbf v = (\frac {\sqrt 6}{6} \mathbf i + \frac {\sqrt 6}{6} \mathbf j -\frac {\sqrt 6}{3} \mathbf k)$
$(x,y,z) = (1,1,-1) + 3\mathbf u \cos t + 3\mathbf v \sin t\\ 
x = 1 + 3\frac {\sqrt 2}{2} \cos t + \frac{\sqrt {6}}{2} \sin t\\
y = 1 - 3\frac {\sqrt 2}{2} \cos t + \frac {\sqrt {6}}{2} \sin t\\
z = -1  - \sqrt {6} \sin t$
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