Answer
Answer: Since the vectors are not unit vectors the set is not orthonormal.
Normalized vector $u_{2}$: <1/$\sqrt {(5)}$, 2/$\sqrt {(5)}$, 0>
Work Step by Step
Note: Vectors can be rewritten using <>, so
$u_{1}$ =
$u_{2}$ = <1/3, 2/3, 0>
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.
$u_{1}$∙$u_{2}$ = ∙<1/3, 2/3,0> = (-2/3)(1/3) + (1/3)(2/3) + (2/3)(0) = (-2/9) + (2/9) + (0) = 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.
||$u_{1}$|| = $\sqrt {(-2/3)^{2} + (1/3)^{2} + (2/3)^{2}}$ = $\sqrt {1}$ = 1
||$u_{2}$|| = $\sqrt {(1/3)^{2} + (2/3)^{2} + (0)^{2}}$ = $\sqrt {(5/9)}$ = $\sqrt {(5)}$/3
Since the magnitudes of both $u_{1}$ and $u_{2}$ are not equal to 1, both of the vectors are not unit vectors.
Answer: Since the vectors are not unit vectors the set is not orthonormal.
3. We now have to normalize the orthogonal set to be orthonormal. We can do this by dividing each component of the vectors by its respective magnitude.
$u_{1}$: This is already a unit vector since the magnitude equals 1.
$u_{2}$:
1st Component: $\frac{1/3}{\sqrt {(5)}/3}$ = 1/$\sqrt {(5)}$
2nd Component: $\frac{2/3}{\sqrt {(5)}/3}$ = 2/$\sqrt {(5)}$
3rd Component: $\frac{0}{\sqrt {(5)}/3}$ = 0
Normalized vector $u_{2}$: <1/$\sqrt {(5)}$, 2/$\sqrt {(5)}$, 0>