The gear reducer is subjected to the couple moments shown. Determine the resultant couple moment and specify its magnitude and coordinate direction angles.

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.
Solution:
Show me the final answer↓
Let us first express each moment acting upon the gear reducer in Cartesian vector form.
M_1=\left\{0i+50j+0k\right\}\,\text{N}\cdot\text{m}
M_2=\left\{60\cos30^0i+0j+60\sin30^0k\right\}\,\text{N}\cdot\text{m}
M_2=\left\{51.96i+0j+30k\right\}\,\text{N}\cdot\text{m}
M_2=\left\{60\cos30^0i+0j+60\sin30^0k\right\}\,\text{N}\cdot\text{m}
M_2=\left\{51.96i+0j+30k\right\}\,\text{N}\cdot\text{m}
The resultant couple moment can be found by adding the two moments together.
M_c=M_1+M_2
M_c=\left\{0i+50j+0k\right\}+\left\{51.96i+0j+30k\right\}
M_c=\left\{51.96i+50j+30k\right\}\,\text{N}\cdot\text{m}
M_c=\left\{0i+50j+0k\right\}+\left\{51.96i+0j+30k\right\}
M_c=\left\{51.96i+50j+30k\right\}\,\text{N}\cdot\text{m}
The magnitude of this resultant moment is:
M_c=\sqrt{(51.96)^2+(50)^2+(30)^2}
M_c=78.1\,\text{N}\cdot\text{m}
M_c=78.1\,\text{N}\cdot\text{m}
The magnitude is equal to the square root of the sum of the squares of the vector. If the position vector was r\,=\,ai+bj+ck, then the magnitude would be, r_{magnitude}\,=\,\sqrt{(a^2)+(b^2)+(c^2)}. In the simplest sense, you take each term of a vector, square it, add it together, and then take the square root of that value.
We can also calculate the coordinate direction angles as follows:
\alpha=\cos^{-1}\left(\dfrac{51.96}{78.1}\right)=48.29^0
\beta=\cos^{-1}\left(\dfrac{50}{78.1}\right)=50.19^0
\gamma=\cos^{-1}\left(\dfrac{30}{78.1}\right)=67.41^0
\beta=\cos^{-1}\left(\dfrac{50}{78.1}\right)=50.19^0
\gamma=\cos^{-1}\left(\dfrac{30}{78.1}\right)=67.41^0
Final Answers:
M_c=\left\{51.96i+50j+30k\right\}\,\text{N}\cdot\text{m}
Magnitude of M_c=78.1\,\text{N}\cdot\text{m}
Magnitude of M_c=78.1\,\text{N}\cdot\text{m}
\alpha=48.29^0
\beta=50.19^0
\gamma=67.41^0