Given an orthonormal positively oriented basis $B=(\mathbf{i}, \mathbf{j}, \mathbf{k})$ and a real number $h$, let $$ \mathbf{v} = \mathbf{i}-h\mathbf{k} $$ $$ \mathbf{w} = 2\mathbf{i}-\mathbf{j}+\mathbf{k} $$ $$ \mathbf{z}= h\mathbf{i}+\mathbf{j}-\mathbf{k} $$ Determine for which values of $h$ we have that $(\mathbf{v},\mathbf{w},\mathbf{z})$ is a basis and it is a positively oriented basis.
I found that for $h\neq0$ and $h\neq-2$, $(\mathbf{v},\mathbf{w},\mathbf{z})$ is a basis, but I cannot find the values for which $(\mathbf{v},\mathbf{w},\mathbf{z})$ is a positively oriented basis.
(This exercise must be solved without using matrices!)
If you are allowed to use the determinant (it is not a matrix !) with respect to $(i,j,k)$:
$$\begin{vmatrix}1&0&-h\\2&-1&1\\h&1&-1\end{vmatrix} \ = \ -2h-h^2$$
Using the classical result about the sign of a quadratic expression, $(v,w,z)$
is not a basis for $h=-2$ or $h=0$.
has the same orentation as $(i,j,k)$, i.e., is positively oriented basis iff $-2<h<0$.
is a negatively oriented basis in the other cases: $h<-2$ or $h>0$.