Answer
Since all pairs are orthogonal, the set it orthogonal.
Work Step by Step
Note: Vectors can be rewritten using <>, so
$u_{1}$ = <1, -2, 1>
$u_{2}$ = <0, 1, 2>
$u_{3}$ =
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 $u_{2}$.
$u_{1}$∙$u_{2}$ = <1, -2, 1>∙<0, 1, 2> = (1)0 + (-2)1 + (1)2 = 0 + (-2) + 2 = 0
Since $u_{1}$∙$u_{2}$ = 0, the set is orthogonal.
The second pair is $u_{1}$ and $u_{3}$.
$u_{1}$∙$u_{3}$ = <1, -2, 1>∙ = (1)(-5) + (-2)(-2) + (1)(1) = -5 + 4 + 1 = 0
Since $u_{1}$∙$u_{3}$ = 0, the set is orthogonal.
The third pair is $u_{2}$ and $u_{3}$.
$u_{2}$∙$u_{3}$ = <0, 1, 2>∙ = (0)(-5) + (1)(-2) + (2)(1) = 0 + (-2) + 2 = 0
Since $u_{2}$∙$u_{3}$ = 0, the set is orthogonal.
Since all pairs are orthogonal, the set it orthogonal.