The question is:
Find, if possible, an orthogonal unit vector at: $2i + 3j - k$ and $-2i - 3j + 4k$.
$$\left|\begin{matrix} i & j & k \\ 2 & 3 & -1 \\ -2 & -3 & 4 \end{matrix}\right| = [9, -6, 0]$$
His norm is $\sqrt{117}$
The orthogonal unit vector is $\dfrac{[ 9, -6, 0 ]}{\sqrt{117}}.$
Your question needs to be more detailed and organized because I am not exactly sure what you want? If you are wanting to know how to get orthonormal vectors then you can use projections. The projection onto a vector x is give by. $$Pv=\frac{xx^T}{||x||^2}v$$