The cross product of two vectors gives a third vector [tex]\mathbf v[/tex] that is orthogonal to the first two.
[tex]\mathbf v=(\vec i+\vec j+\vec k)\times(4\,\vec i+\vec k)=\begin{vmatrix}\vec i&\vec j&\vec k\\1&1&1\\4&0&1\end{vmatrix}=\vec i+3\,\vec j-4\,\vec k[/tex]
Normalize this vector by dividing it by its norm:
[tex]\dfrac{\mathbf v}{\|\mathbf v\|}=\dfrac{\vec i+3\,\vec j-4\,\vec k}{\sqrt{1^2+3^2+(-4)^2}}=\dfrac1{\sqrt{26}}(\vec i+3\,\vec j-4\vec k)[/tex]
To get another vector orthogonal to the first two, you can just change the sign and use [tex]-\mathbf v[/tex].
