Answer
The set $\{(4,-1,1),(-1,0,4),(-4,-17,-1)\}$ is orthogonal, not orthonormal and it is a basis for $R^3$.
Work Step by Step
given $\{(4,-1,1),(-1,0,4),(-4,-17,-1)\}$
let $v_1=(4,-1,1),v_2=(-1,0,4),v_3=(-4,-17,-1)$
(a)
since
$v_1v_2=-4+0+4=0$
$v_1v_3=-16+17-1=0$
$v_2v_3=4+0-4=0$
then, the set $\{(4,-1,1),(-1,0,4),(-4,-17,-1)\}$ is orthogonal.
(b)
scince
$\left\|{v}_{1}\right\|=\sqrt{v_{1} \cdot v_{1}}=\sqrt{16+1+1}=\sqrt{18}\neq1$
then, the set $\{(4,-1,1),(-1,0,4),(-4,-17,-1)\}$ is not orthonormal.
(c)
by the corollary to Theorem 5.10, $\{(4,-1,1),(-1,0,4),(-4,-17,-1)\}$ is a basis for $R^3$.
