Let an $x´,y´,z´$ coordinate system be obtained by rotating an xyz-coordinate system about the z-axis through an angle of $-45^\circ$. if i , j and k are the standard unit vectors in $xyz$-system the components of these vectors in the $x´,y´,z´$-system are:
i$=(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ j=$(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ k=$(0,0,1)$
What is the relationship between $\sqrt{2}$ and $45^\circ$?
Why divide 1 by $\sqrt{2}$?
Thanks in advance!
It's the cosine and sine of the angle:
$$ \cos(45^\circ) = \sin(45^\circ) = \frac{1}{\sqrt{2}} $$
This may be more evident if you convert to radians, $ 45^\circ = \frac{\pi}{4} \text{ rad}$.