Maximizing angle between three unit vectors

Question

Find the maximum angle between three equally inclined unit vectors in the 3D space.

My attempt

Define unit vectors $\hat{a},\hat{b},\hat{c}$ and their components $a_i,b_i,c_i, i\in \{1,2,3\}$

Note that $a_1b_1+a_2b_2+a_3b_3=c_1b_1+c_2b_2+c_3b_3=a_1c_1+a_2c_2+a_3c_3$

So maximize

$$\arccos(a_1b_1+a_2b_2+a_3b_3) \text{ or } \arccos(c_1b_1+c_2b_2+c_3b_3) \text{ or } \arccos(a_1c_1+a_2c_2+a_3c_3)$$

under the constraint

$$\sqrt{a_1^2+a_2^2+a_3^2}=\sqrt{b_1^2+b_2^2+b_3^2}=\sqrt{c_1^2+c_2^2+c_3^2}=1$$

I'm not sure what to do next. Any help is appreciated!

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Update

WLOG it can be assumed that any one of the vectors is $\hat{i}$ or $\hat{j}$ or $\hat{k}$

This would reduce the number of variables to 6

Answer 1

Lets not complicate it...

Let $p,q,r$ be unit the vectors indeed $$|p+q+r|^2\ge 0$$

$$\implies 3+2(p\cdot q+q\cdot r+r\cdot p)\ge 0$$ $$3+6\cos \theta\ge 0$$ Can you end it now?