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
$u_{1}$ = <5, -4, 0, 3>
$u_{2}$ =
$u_{3}$ = <3, 3, 5, -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 $u_{2}$.
$u_{1}$∙$u_{2}$ = <5, -4, 0, 3>∙ = (5)(-4) + (-4)(1) + (0)(-3) + (3)(8) = -20 + (-4) + 0 + 24 = 0
Since $u_{1}$∙$u_{2}$ = 0, the pair is orthogonal.
The second pair is $u_{1}$ and $u_{3}$.
$u_{1}$∙$u_{3}$ = <5, -4, 0, 3>∙<3, 3, 5, -1> = (5)(3) + (-4)(3) + (0)(5) + (3)(-1) = 15 + (-12) + 0 + (-3) = 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, 3, 5, -1> = (-4)(3) + (1)(3) + (-3)(5) + (8)(-1) = -12 + 3 + (-15) + (-8) = -32
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.