If a, b and c are mutually perpendicular vectors of equal magnitude, then prove that the vector (2 a + b +2 c ) is equally inclined to both a and c.…

Let, \(\displaystyle |\vec{a}|=|\vec{b}|=|\vec{c}|=m\) \[\begin{aligned} |2 \vec{a}+\vec{b}+2 \vec{c}|^{2} & =4|\vec{a}|^{2}+|\vec{b}|^{2}+4|\vec{c}|^{2}+4 \vec{a} \cdot \vec{b}+4 \vec{b} \cdot \vec{c}+8 \vec{a} \cdot \vec{c} \\ & =9 m^{2} \\ |2 \vec{a}+\vec{b}+2 \vec{c}| & =3 m \end{aligned} \] Let \(\displaystyle \theta\) be angle between \(\displaystyle 2 \vec{a}+\vec{b}+2 \vec{c}\) and \(\displaystyle \vec{a}\). \(\displaystyle \cos \theta=\frac{(2 \vec{a}+\vec{b}+2 \vec{c}) \cdot \vec{a}}{|2 \vec{a}+\vec{b}+2 \vec{c}||\vec{a}|}=\frac{2 m^{2}}{3 m \cdot m}=\frac{2}{3}\) Let, \(\displaystyle \alpha\) be angle between \(\displaystyle 2 \vec{a}+\vec{b}+2 \vec{c}\) and \(\displaystyle \vec{c}\)