Answer
Since the pair $u_{2}$ and $u_{3}$ is not orthogonal, the set is not orthogonal.
Work Step by Step
Note: Vectors can be rewritten using <>, so
$u_{1}$ = <2, -7, -1>
$u_{2}$ =
$u_{3}$ = <3, 1, -1>
To determine if the set is orthogonal, we determine if each pair of distinct vectors is orthogonal. If all the pairs are, then the set is orthogonal.
Note: To determine if a pair is orthogonal, we take the dot product, and see if the dot product is equal to 0.
The first pair is $u_{1}$ and $u_{2}$.
$u_{1}$∙$u_{2}$ = <2, -7, -1>∙ = (2)(-6) + (-7)(-3) + (-1)9 = -12 + 21 + -9 = 0
Since $u_{1}$∙$u_{2}$ = 0, the pair is orthogonal.
The second pair is $u_{1}$ and $u_{3}$.
$u_{1}$∙$u_{3}$ = <2, -7, -1>∙<3, 1, -1> = (2)(3) + (-7)(1) + (-1)(-1) = 6 + (-7) + 1 = 0
Since $u_{1}$∙$u_{3}$ = 0, the pair is orthogonal.
The third pair is $u_{2}$ and $u_{3}$.
$u_{2}$∙$u_{3}$ = ∙<3, 1, -1> = (-6)(3) + (-3)(1) + (9)(-1) = -18 + (-3) + (-9) = -30
Since $u_{2}$∙$u_{3}$ $\ne$ 0, the pair is not orthogonal.
Since the pair $u_{2}$ and $u_{3}$ is not orthogonal, the set is not orthogonal.