A set of vectors is orthonormal if each vector is a unit vector ( length or norm is equal to \( 1\)) and all vectors in the set are orthogonal to each other. Therefore a basis is orthonormal if the set of vectors in the basis is orthonormal. The vectors in a set of orthogonal vectors are linearly independent.

Example 1

Show that the vectors \( \textbf i = \begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix} \) ,
\( \textbf j = \begin{bmatrix}
0\\
1\\
0
\end{bmatrix} \) and
\( \textbf k = \begin{bmatrix}
0\\
0\\
1
\end{bmatrix} \) form an ortonormal basis.

Solution to Example 1

Show that the inner product of each pair of vectors is equal to 0.

\( \textbf i \cdot \textbf j = (1)(0)+(0)(1)+(0)(0) = 0 \)

\( \textbf i \cdot \textbf k = (1)(0)+(0)(0)+(0)(1) = 0 \)

\( \textbf j \cdot \textbf k = (0)(0)+(1)(0)+(0)(1) = 0 \)

Therefore the three vectors are orthogonal and also linearly independent.

Show that the length or norm of each vector is equal to 1.

\( ||i|| = \sqrt {1^2+0^2+0^2} = \sqrt 1 = 1 \)

\( ||j|| = \sqrt {0^2+ 1^2+0^2} = \sqrt 1 = 1 \)

\( ||j|| = \sqrt {0^2+ 0^2+1^2} = \sqrt 1 = 1 \)

The three vectors are orthogonal to each other and the length of each is equal to 1; they therefore form an orthonormal basis.

Example 2

a) Show that the vectors \( \; \textbf u = K \begin{bmatrix}
1\\
0 \\
2
\end{bmatrix} \) ,
\( \; \textbf v = L \begin{bmatrix}
4 \\
1 \\
-2
\end{bmatrix} \) and \( \; \textbf w = M \begin{bmatrix}
-2 \\
10 \\
1
\end{bmatrix} \), where \( K \), \( L \) and \( M \) are real constants, are orthogonal.

b) Find positive values for the constants \( K \), \( L \) and \( M \) so that the vectors \( \textbf u \) , \( \textbf v \) and \( \textbf w \) form an orthonormal basis.

Solution to Example 2

a)

Calculate the inner product of each pair of vectors.

\( \textbf u \cdot \textbf v = KL(1)(4)+KL(0)(1)+KL(2)(-2) = 4 KL - 4 KL = 0 \)

\( \textbf u \cdot \textbf w = KM(1)(-2)+KM(0)(10)+KM(2)(1) = -2 KM + 2 KM = 0 \)

\( \textbf v \cdot \textbf w = LM(4)(-2)+LM(1)(10)+LM(-2)(1) = -8LM + 10 LM - 2LM = 0 \)

Therefore vectors \( \textbf u \), \( \textbf v \) and \( \textbf w \) are orthogonal and linearly independent.

b)

We now need to find the length (or norm) of each vectors and set it equal to \( 1 \).

\( ||\textbf u|| = \sqrt {k^2 + 0^2 + (2k)^2} = 1 \)

which gives the equation

\( \sqrt {5 k^2} = 1 \)

Square both sides of the equation above and solve for \( K \) positive.

\( 5 k^2 = 1 \) gives \( K = \dfrac{1}{\sqrt 5} \)

\( ||\textbf v|| = \sqrt {(4L)^2 + L^2 + (-2L)^2} = 1 \)

which gives the equation

\( \sqrt {21 L^2} = 1 \)

Square both sides of the equation above and solve for \( L \) positive.

\( 21 L^2 = 1 \) gives \( L = \dfrac{1}{\sqrt {21}} \)

\( ||\textbf w|| = \sqrt {(-2M)^2 + (10M)^2 + M^2} = 1 \)

which gives the equation

\( \sqrt {105 M^2} = 1 \)

Square both sides of the equation above and solve for \( M \) positive.

\( 105 M^2 = 1 \) gives \( M = \dfrac{1}{\sqrt {105}} \)

Example 3

Proove that the vectors \( \textbf u =
\begin{bmatrix}
\sin(\theta) \cos (\phi) \\
\sin(\theta) \sin (\phi) \\
\cos (\theta)
\end{bmatrix}
\) ,
\( \textbf v =
\begin{bmatrix}
\cos(\theta) \cos (\phi) \\
\cos(\theta) \sin (\phi) \\
-\sin (\theta)
\end{bmatrix}
\) and
\( \textbf w =
\begin{bmatrix}
-\sin (\phi) \\
\cos (\phi) \\
0
\end{bmatrix}
\) form an orthonormal basis.

Solution to Example 3

In example 6 of orthogonal vectors, it has been prooved that vectors \( \textbf u \), \( \textbf v \) and \( \textbf w \) given above are orthogonal and therefore linearly independent.

We now need to show that the norm of each vector is equal to \( 1 \).

\( ||\textbf u || = \sqrt { (\sin(\theta) \cos (\phi))^2 +(\sin(\theta) \sin (\phi))^2 +(\cos (\theta))^2 } \)

Expand

\( ||\textbf u || = \sqrt { \sin^2(\theta) \cos^2 (\phi) + \sin^2(\theta) \sin^2 (\phi) +\cos^2 (\theta)) } \)

Factor \( \sin^2(\theta) \) out from the first and second term of the radicand

\( ||\textbf u || = \sqrt { \sin^2(\theta) [ \cos^2 (\phi) + \sin^2 (\phi) ] +\cos^2 (\theta)) } \)

Use trigonometric identity \( \cos^2 (\phi) + \sin^2 (\phi) = 1\) to simplify the above to

\( ||\textbf u || = \sqrt { \sin^2(\theta) +\cos^2 (\theta)) } \)

Use trigonometric identity \( \cos^2 (\theta) + \sin^2 (\theta) = 1\) to simplify the above to

\( ||\textbf u || = \sqrt 1 = 1 \)

\( ||\textbf v || = \sqrt { (\cos(\theta) \cos (\phi))^2 +(\cos(\theta) \sin (\phi))^2 +(-\sin (\theta))^2 } \)

Expand

\( ||\textbf v || = \sqrt { \cos^2(\theta) \cos^2 (\phi) + \cos^2(\theta) \sin^2 (\phi) +\sin^2 (\theta)) } \)

Factor \( \cos^2(\theta) \) out from the first and second term of the radicand

\( ||\textbf v || = \sqrt { \cos^2(\theta) [ \cos^2 (\phi) + \sin^2 (\phi) ] +\sin^2 (\theta)) } \)

Use trigonometric identity \( \cos^2 (\phi) + \sin^2 (\phi) = 1\) to simplify the above to

\( ||\textbf v || = \sqrt { \cos^2(\theta) +\sin^2 (\theta)) } \)

Use trigonometric identity \( \cos^2 (\theta) + \sin^2 (\theta) = 1\) to simplify the above to

\( ||\textbf v || = \sqrt 1 = 1 \)

\( ||\textbf w || = \sqrt { (-\sin (\phi))^2 +(\cos (\phi))^2 + 0^2 }\)

Simplify

\( ||\textbf w || = \sqrt { \sin^2 (\phi) + \cos^2 (\phi) } \)

Use trigonometric identity \( \cos^2 (\phi) + \sin^2 (\phi) = 1\) to simplify the above to

\( ||\textbf w || = \sqrt 1 = 1 \)

The vectors \( \textbf u \) , \( \textbf v \) and \( \textbf w \) form an orthonormal basis.

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