Answer
Since the pair $u_{1}$ and $u_{3}$ is not orthogonal, the set of vectors is not orthogonal.
Work Step by Step
Note: Vectors can be rewritten using <>, so
$u_{1}$ =
$u_{2}$ = <5, 2, 1>
$u_{3}$ = <3, -4, -7>
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, 2, 1> = (-1)5 + 4(2) + (-3)1 = -5 + 8 + (-3) = 0
Since $u_{1}$∙$u_{2}$ = 0, the set is orthogonal.
The second pair is $u_{1}$ and $u_{3}$.
$u_{1}$∙$u_{3}$ = ∙<3, -4, -7> = (-1)3 + (4)(-4) + (-3)(-7) = -3 + (-16) + 21 = 2
Since $u_{1}$∙$u_{3}$ $\ne$ 0, the set is not orthogonal.
The third pair is $u_{2}$ and $u_{3}$.
$u_{2}$∙$u_{3}$ = <5, 2, 1>∙<3, -4, -7> = (5)3 + (2)(-4) + (1)(-7) = 15 + (-8) + -7 = 0
Since $u_{2}$∙$u_{3}$ = 0, the set is orthogonal.
Since the pair $u_{1}$ and $u_{3}$ is not orthogonal, the set of vectors is not orthogonal.
