If $\vec a,\overrightarrow b ,\overrightarrow c$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\vec a + \vec b + \vec c$ is equally inclined to $\vec a,\vec b$ and $\vec c$.

Here $|\vec a\vec b| = |\vec c|$ and $\vec a \cdot \vec b = \vec b \cdot \vec c = \vec c \cdot \vec a = 0$

…(i)
Let $\alpha ,\beta ,\gamma$ be the angles which $\vec a + \vec b + \vec c$ makes with $\vec a,\vec b,\vec c$ respectively.

$\therefore$ $\cos \alpha = \cfrac{{\vec a \cdot (\vec a + \vec b + \vec c)}}{{|\vec a||\vec a + \vec b + \vec c|}} = \cfrac{{\vec a \cdot \vec a + \vec a \cdot \vec b + \vec a \cdot \vec c}}{{|\vec a||\vec a + \vec b + \vec c|}}$

$= \cfrac{{|\vec a|}}{{|\vec a + \vec b + \vec c|}}$ [Using (i)]

…(ii)

Similarly, $\cos \beta = \cfrac{{|\vec b|}}{{|\vec a + \vec b + \vec c|}}$

…(iii)
and $\cos \gamma = \cfrac{{|\vec c|}}{{|\vec a + \vec b + \vec c|}}$

…(iv)
From (ii), (iii) and (iv),

we get $\cos \alpha = \cos \beta = \cos \gamma$

$\Rightarrow \alpha = \beta = \gamma$
Hence the vector $(\vec a + \vec b + \vec c)$ is equally inclined to $\vec a,\vec b$ and ${\vec c_:}$