Answer
Since all of the pairs are orthogonal, the set is orthogonal.
Work Step by Step
Note: Vectors can be rewritten using <>, so
$u_{1}$ = <3, -2, 1, 3>
$u_{2}$ =
$u_{3}$ = <3, 8, 7, 0>
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 $u_{2}$.
$u_{1}$∙$u_{2}$ = <3, -2, 1, 3>∙ = (3)(-1) + (-2)(3) + (1)(-3) + (3)(4) = -3 + (-6) + (-3) + 12 = 0
Since $u_{1}$∙$u_{2}$ = 0, the pair is orthogonal.
The second pair is $u_{1}$ and $u_{3}$.
$u_{1}$∙$u_{3}$ = <3, -2, 1, 3>∙<3, 8, 7, 0> = (3)(3) + (-2)(8) + (1)(7) + (3)0 = 9 + (-16) + 7 + 0 = 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, 8, 7, 0> = (-1)(3) + (3)(8) + (-3)(7) + (4)0 = -3 + 24 + (-21) + 0 = 0
Since $u_{2}$∙$u_{3}$ = 0, the pair is orthogonal.
Since all of the pairs are orthogonal, the set is orthogonal.
