The matrix below shows $6$ entries of a $33$ matrix where the

values of the final column are denoted by the variable $A,B,$ and

C$.$

$R=\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& A\\ \frac{1}{2}& \frac{1}{2}& B\\ \frac{1}{\sqrt[\phantom{2}]{2}}& -\frac{1}{\sqrt[\phantom{2}]{2}}& C\end{array}\right]$

Given the above, demonstrate which of the two column vectors

from the options below will ensure that $R$ is a rotation matrix.

$($i$)\left[\begin{array}{c}A\\ B\\ C\end{array}\right]=\left[\begin{array}{c}-\frac{1}{\sqrt[\phantom{2}]{2}}\\ \frac{1}{\sqrt[\phantom{2}]{2}}\\ 0\end{array}\right]$

$($ii$)\left[\begin{array}{c}A\\ B\\ C\end{array}\right]=\left[\begin{array}{c}0\\ \frac{1}{\sqrt[\phantom{2}]{2}}\\ -\frac{1}{\sqrt[\phantom{2}]{2}}\end{array}\right]$