Answer
Answer: Since the vectors are orthogonal and are unit vectors, the set is orthonormal.
Work Step by Step
Note: Vectors can be rewritten using <>, so
$u_{1}$ =
$u_{2}$ =
A set of vectors is orthonormal if the set is orthogonal and if each of the vectors is a unit vector. To determine if the set is orthonormal we must first determine if the set is orthogonal, and then we need to determine if each vector is a unit vector.
1. Determine if the set of vectors is orthogonal.
Note: To determine if a set is orthogonal, each pair of vectors must be orthogonal meaning the dot product = 0.
In this case, $u_{1}$∙$u_{2}$ must equal 0 to be orthogonal.
$u_{1}$∙$u_{2}$ = ∙ = (-.6)(.8) + (.8)(.6) = (-.48) + .48 = 0
Since $u_{1}$∙$u_{2}$ = 0, the pair is orthogonal. Since there is only one pair, the set is orthogonal.
2. Determine if each of the vectors are unit vectors.
Note: To determine if a vector is a unit vector, we take the magnitude of each vector. Unit vectors have a magnitude of 1.
||$u_{1}$|| = $\sqrt {(-.6)^{2} + (.8)^{2} }$ = $\sqrt {1}$ = 1
||$u_{2}$|| = $\sqrt {(.8)^{2} + (.6)^{2} }$ = $\sqrt {1}$ = 1
Since the magnitudes of both $u_{1}$ and $u_{2}$ are equal to 1, both of the vectors are not unit vectors.
Answer: Since the vectors are orthogonal and are unit vectors, the set is orthonormal.
