Given the orthonormal ordered base pairs $ \hat s_1, \hat s_2$ of $V = R ^ 3 $ below, determine if $\hat s_1, \hat s_2 $ has the same orientation or has opposite orientations.
a-) $ \hat s_1$=($\vec e_1 =(0,0,1), \vec e_2 =(1,0,0), \vec e_3$ =(0,1,0))
$ \hat s_2$=( $\vec e'_1 =(\frac{1}{\sqrt 2},0,-\frac{1}{\sqrt 2}), \vec e'_2 =(\frac{1}{\sqrt 3},\frac{1}{\sqrt 3},\frac{1}{\sqrt 3}), \vec e'_3 =(\frac{1}{\sqrt 6},-\frac{2}{\sqrt 6},\frac{1}{\sqrt 6}))$
b-)$ \hat s_1$=($\vec e_1 =(\frac{2}{3},\frac{1}{3},\frac{2}{3}), \vec e_2 =(\frac{1}{\sqrt 18},-\frac{4}{\sqrt 18},\frac{1}{\sqrt 18}), \vec e_3 =(\frac{3}{\sqrt 19},-\frac{1}{\sqrt 19},\frac{3}{\sqrt 19}))$
$ \hat s_2$= ( $\vec e'_1 =(0,\frac{1}{\sqrt 2},\frac{1}{\sqrt 2}), \vec e'_2 =(1,0,0), \vec e'_3 =(0,\frac{1}{\sqrt 2},-\frac{1}{\sqrt 2}))$
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I did like that
a-)$(\frac{1}{\sqrt 2}, 0,\frac{1}{\sqrt 2})$= a(1,0,0) + b(1,0,0) + c(0,0,1)
a=$\frac{1}{\sqrt 2}$, b=0, c=$-\frac{1}{\sqrt 2}$
$(\frac{1}{\sqrt 3}, \frac{1}{\sqrt 3},\frac{1}{\sqrt 3})$= d(1,0,0) + e(1,0,0) + f(0,0,1)
d=e=f= $\frac{1}{\sqrt 3}$
$(\frac{1}{\sqrt 6}, -\frac{2}{\sqrt 6},\frac{1}{\sqrt 6})$= g(1,0,0) + h(1,0,0) + i(0,0,1)
g=$\frac{1}{\sqrt 6}, h=-\frac{2}{\sqrt 6}, i=\frac{1}{\sqrt 6}$
det $\begin{bmatrix}a&b&c\\ d&e&f\\ g&h&i\\ \end{bmatrix}=1$
det = 1 same orientation
b-)
$(\frac{2}{3},\frac{1}{3},\frac{2}{3})$= a(0,$\frac{1}{\sqrt 2}$,$\frac{1}{\sqrt 2}$) + b(1,0,0) + c(0,$\frac{1}{\sqrt 2}$,-$\frac{1}{\sqrt 2}$)
a=$\frac{1}{\sqrt 2}$, b=$\frac{2}{3}$, c=-$\frac{1}{3\sqrt 2}$
$(\frac{1}{\sqrt 18}, -\frac{4}{\sqrt 18},\frac{1}{\sqrt 18})$= d(0,$\frac{1}{\sqrt 2}$,$\frac{1}{\sqrt 2}$) + e(1,0,0) + f(0,$\frac{1}{\sqrt 2}$,-$\frac{1}{\sqrt 2}$)
d=$-\frac{1}{2}, e=\frac{1}{3\sqrt 2}, f=-\frac{5}{6}$
$(\frac{3}{\sqrt 19}, -\frac{1}{\sqrt 19},\frac{3}{\sqrt 19})$= g(0,$\frac{1}{\sqrt 2}$,$\frac{1}{\sqrt 2}$) + h(1,0,0) + i(0,$\frac{1}{\sqrt 2}$,-$\frac{1}{\sqrt 2}$)
g=$\sqrt\frac{2}{19}, h= \frac{3}{\sqrt 19}, i=-2\sqrt\frac{2}{19}$
det $\begin{bmatrix}a&b&c\\ d&e&f\\ g&h&i\\ \end{bmatrix}=\frac{\sqrt 38}{228}>0$ same orientation?