I've just started studying vector algebra so go easy on me. I'm not sure how to approach this problem.
"The vector OP makes an angle of 60 degrees with the positive x-axis and 45 degrees with the positive y-axis. Find the possible angles that the vector can make with the z-axis."
I'm self-studying and the question is from a textbook with answers but no working shown. I believe I could use the dot or cross product to solve but that hasn't been covered yet in the textbook so I think there must be another way to do it....
You can use the concept of directional cosines, employing the identity:
$\cos^2 \theta_x + \cos^2 \theta_y + \cos^2 \theta_z = 1$
Inputting values we get: $$ \left(\frac{1}{2}\right)^2 + \left(\frac{1}{\sqrt 2}\right)^2 +\cos^2 \theta_z = 1$$
$\cos^2 \theta_z= \frac{1}{4}$
$\cos \theta_z = \pm \frac{1}{2}$