Answer
Since all of the pairs are orthogonal, the set is orthogonal.
Work Step by Step
Note: Vectors can be rewritten using <>, so
$u_{1}$ = <2, -5, -3>
$u_{2}$ = <0, 0, 0>
$u_{3}$ = <4, -2, 6>
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, -5, -3>∙<0, 0, 0> = (2)(0) + (-5)(0) + (-3)0 = 0 + 0 + 0 = 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, -5, -3>∙<4, -2, 6> = (2)(4) + (-5)(-2) + (-3)(6) = 8 + 10 + (-18) = 0
Since $u_{1}$∙$u_{3}$ = 0, the pair is orthogonal.
The third pair is $u_{2}$ and $u_{3}$.
$u_{2}$∙$u_{3}$ = <0, 0, 0>∙<4, -2, 6> = (0)(4) + (0)(-2) + (0)(6) = 0 + 0 + 0 = 0
Since $u_{2}$∙$u_{3}$ = 0, the pair is orthogonal.
Since all of the pairs are orthogonal, the set is orthogonal.