Consider a "new" coordinate system $O \bar{x_{1}}\bar{x_{2}}\bar{x_{3}}$ that is obtained by a counter clockwise rotation of $\frac{\pi}{2}$ radians about an axis that coincides with the unit vector $$\underline{n} = \frac{1}{\sqrt{2}}(\underline{e_{1}} + \underline{e_{2}} )$$
in the "old" coordinate system $Ox_{1}x_{2}x_{3}$ as indicated in the sketch below
if $\beta_{ij} = cos(\bar{x_{i}},x_{j})$ (angle between the 2 vectors) calculate $\beta_{ij}$
what I had attempted what I tried creating a new basis and rotating that(so first basis white, second red,third blue) the comparing the blue axis to the white. but I keep getting zero when I do my calculations
could someone assist, please
Let’s denote by $R=(e_1’,e_2’,e_3’)$ the image of $e_1,e_2,e_3)$. One has
$$R=\begin{bmatrix} {1\over 2} & {1\over 2} & {1\over\sqrt{2}}\\ {1\over 2} & {1\over 2} & -{1\over\sqrt{2}}\\ - {1\over\sqrt{2}} & {1\over\sqrt{2}} & 0 \end{bmatrix}$$