Suppose that $a,b$ are two orthogonal unit vectors in $\mathbb R^3$, want to find a unit vector $c$ orthogonal to both $a$ and $b$. And the matrix formed by using $a$, $b$, $c$ as row vectors has determinant 1.
One strategy would be to suppose that $c = (x_1, x_2, x_3)$, and write down three equations using given conditions. Then solve it. But this would be tedious. Are there any other methods?
Best Answer
Take their cross product $a \times b$. Since $a$ and $b$ both have magnitude $1$ and are orthogonal, the result is also a unit vector.