<back
parsing from text
\[\bar{U} \underline U =\\
U_0^2 \left[
\sin\left(\omega t\right) + \cos\left(\omega t\right) i + \sin\left(n \omega t\right) j + \cos\left(n \omega t\right) k
\right] \left[
\sin\left(\omega t\right) – \cos\left(\omega t\right) i – \sin\left(n \omega t\right) j – \cos\left(n \omega t\right) k
\right]\]
distributing factors
\[\bar{U} \underline U =\\
\left[
U_0^2 \sin\left(\omega t\right) \left[
\sin\left(\omega t\right) – \cos\left(\omega t\right) i – \sin\left(n \omega t\right) j – \cos\left(n \omega t\right) k
\right] + U_0^2 \cos\left(\omega t\right) i \left[
\sin\left(\omega t\right) – \cos\left(\omega t\right) i – \sin\left(n \omega t\right) j – \cos\left(n \omega t\right) k
\right] + U_0^2 \sin\left(n \omega t\right) j \left[
\sin\left(\omega t\right) – \cos\left(\omega t\right) i – \sin\left(n \omega t\right) j – \cos\left(n \omega t\right) k
\right] + U_0^2 \cos\left(n \omega t\right) k \left[
\sin\left(\omega t\right) – \cos\left(\omega t\right) i – \sin\left(n \omega t\right) j – \cos\left(n \omega t\right) k
\right]
\right]\]
sorting terms
\[\underline U \bar{U} =\\
\left[
U_0^2 \left[
– \cos\left(\omega n t\right) k – \cos\left(\omega t\right) i – \sin\left(\omega n t\right) j + \sin\left(\omega t\right)
\right] \cos\left(\omega n t\right) k + U_0^2 \left[
– \cos\left(\omega n t\right) k – \cos\left(\omega t\right) i – \sin\left(\omega n t\right) j + \sin\left(\omega t\right)
\right] \cos\left(\omega t\right) i + U_0^2 \left[
– \cos\left(\omega n t\right) k – \cos\left(\omega t\right) i – \sin\left(\omega n t\right) j + \sin\left(\omega t\right)
\right] \sin\left(\omega n t\right) j + U_0^2 \left[
– \cos\left(\omega n t\right) k – \cos\left(\omega t\right) i – \sin\left(\omega n t\right) j + \sin\left(\omega t\right)
\right] \sin\left(\omega t\right)
\right]\]
distributing factors
\[\underline U \bar{U} =\\
\left[
– U_0^2 \cos\left(\omega n t\right) k \cos\left(\omega n t\right) k – U_0^2 \cos\left(\omega t\right) i \cos\left(\omega n t\right) k – U_0^2 \sin\left(\omega n t\right) j \cos\left(\omega n t\right) k + U_0^2 \sin\left(\omega t\right) \cos\left(\omega n t\right) k – U_0^2 \cos\left(\omega n t\right) k \cos\left(\omega t\right) i – U_0^2 \cos\left(\omega t\right) i \cos\left(\omega t\right) i – U_0^2 \sin\left(\omega n t\right) j \cos\left(\omega t\right) i + U_0^2 \sin\left(\omega t\right) \cos\left(\omega t\right) i – U_0^2 \cos\left(\omega n t\right) k \sin\left(\omega n t\right) j – U_0^2 \cos\left(\omega t\right) i \sin\left(\omega n t\right) j – U_0^2 \sin\left(\omega n t\right) j \sin\left(\omega n t\right) j + U_0^2 \sin\left(\omega t\right) \sin\left(\omega n t\right) j – U_0^2 \cos\left(\omega n t\right) k \sin\left(\omega t\right) – U_0^2 \cos\left(\omega t\right) i \sin\left(\omega t\right) – U_0^2 \sin\left(\omega n t\right) j \sin\left(\omega t\right) + U_0^2 \sin\left(\omega t\right) \sin\left(\omega t\right)
\right]\]
sorting terms
\[\underline U \bar{U} =\\
\left[
U_0^2 \cos\left(\omega n t\right) \cos\left(\omega n t\right) – U_0^2 \cos\left(\omega n t\right) \cos\left(\omega t\right) j + U_0^2 \cos\left(\omega n t\right) \cos\left(\omega t\right) j + U_0^2 \cos\left(\omega t\right) \cos\left(\omega t\right) – U_0^2 \sin\left(\omega n t\right) \cos\left(\omega n t\right) i + U_0^2 \sin\left(\omega n t\right) \cos\left(\omega n t\right) i – U_0^2 \sin\left(\omega n t\right) \cos\left(\omega t\right) k + U_0^2 \sin\left(\omega n t\right) \cos\left(\omega t\right) k + U_0^2 \sin\left(\omega n t\right) \sin\left(\omega n t\right) – U_0^2 \sin\left(\omega n t\right) \sin\left(\omega t\right) j + U_0^2 \sin\left(\omega n t\right) \sin\left(\omega t\right) j – U_0^2 \sin\left(\omega t\right) \cos\left(\omega n t\right) k + U_0^2 \sin\left(\omega t\right) \cos\left(\omega n t\right) k – U_0^2 \sin\left(\omega t\right) \cos\left(\omega t\right) i + U_0^2 \sin\left(\omega t\right) \cos\left(\omega t\right) i + U_0^2 \sin\left(\omega t\right) \sin\left(\omega t\right)
\right]\]
collecting similar terms
\[\underline U \bar{U} =\\
\left[
U_0^2 \cos^2\left(\omega n t\right) + U_0^2 \cos^2\left(\omega t\right) + U_0^2 \sin^2\left(\omega n t\right) + U_0^2 \sin^2\left(\omega t\right)
\right]\]
expanding for double angles
\[\underline U \bar{U} =\\
\left[
U_0^2 \left[
\frac{\left[
1
\right]}{\left[
2
\right]} + \frac{\left[
1
\right]}{\left[
2
\right]} \cos\left(2 \omega n t\right)
\right] + U_0^2 \left[
\frac{\left[
1
\right]}{\left[
2
\right]} + \frac{\left[
1
\right]}{\left[
2
\right]} \cos\left(2 \omega t\right)
\right] + U_0^2 \left[
\frac{\left[
1
\right]}{\left[
2
\right]} – \frac{\left[
1
\right]}{\left[
2
\right]} \cos\left(2 \omega n t\right)
\right] + U_0^2 \left[
\frac{\left[
1
\right]}{\left[
2
\right]} – \frac{\left[
1
\right]}{\left[
2
\right]} \cos\left(2 \omega t\right)
\right]
\right]\]
distributing factors
\[\underline U \bar{U} =\\
\left[
U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} + U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} \cos\left(2 \omega n t\right) + U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} + U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} \cos\left(2 \omega t\right) + U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} – U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} \cos\left(2 \omega n t\right) + U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} – U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} \cos\left(2 \omega t\right)
\right]\]
distributing factors
\[\underline U \bar{U} =\\
\left[
U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} + U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} \cos\left(2 \omega n t\right) + U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} + U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} \cos\left(2 \omega t\right) + U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} – U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} \cos\left(2 \omega n t\right) + U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} – U_0^2 \frac{\left[
1
\right]}{\left[
2
\right]} \cos\left(2 \omega t\right)
\right]\]
collecting similar terms
\[\underline U \bar{U} = 2 U_0^2\]