Finding a unit vector orthogonal to vectors $a$ and $b$

Question

If I understand correctly, the cross product of vectors $a$ and $b$ is orthogonal to both $a$ and $b$.

So for an assignment I have to find two unit vectors orthogonal to vector $a = \langle 1,0,4 \rangle$ and $b = \langle 1,-4,2 \rangle$.

The cross product of those two vectors is according to my calculation $\langle 16,2,-4 \rangle$.

Now I have to calculate the unit vector of the cross product, which is according to my calculation $\displaystyle \frac{1}{\sqrt{264}} \langle 16,2,-4 \rangle$, but apparently this is incorrect as the answer must be, according to mybook, $\displaystyle \frac{1}{\sqrt{69}} \langle 8,1,-2 \rangle$ or the same thing but negative?

Answer 1

You made a calculation error while finding the length of the vector <16,2,-4>. The length is $\sqrt {276}$, not $\sqrt {264}$. If you correct that, you will find that your answer, after simplification, is the same as the book's. (I have corrected an error pointed out by user84413 and added detail suggested by Jean-Claude Arbaut: thanks to both.)