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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 $\dfrac{[ 9, -6, 0 ]}{\sqrt{117}}.$

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I assume you are using the cross product to obtain $(9, -6, 0)$ which is orthogonal to $(2, 3, -1)$ and $(-2, -3, 4)$. In that case, what you did is correct. – TMM Nov 2 '12 at 20:23
I think you are having problems expressing yourself in a form which will enable you to get help from this forum. You seem to have applied a method which will work, but you haven't asked us a question - are you unsure why this works? Or are you unsure of the method and want to confirm it before tackling some other similar problems? Or .. it is unclear what you want to achieve. – Mark Bennet Nov 2 '12 at 20:31
"orthogonal unit vector", also known as orthonormal – Hawk Nov 2 '12 at 20:33

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. $$Pv=\frac{xx^T}{||x||^2}v$$

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