If $\hat v_1,\hat v_2,\hat v_3$ are unit vectors given by
$\hat v_1 = a\hat i + b\hat j + c\hat k$
$\hat v_2 = b\hat i + c\hat j + a\hat k$
$\hat v_3 = c\hat i + a\hat j + b\hat k$
where $a, b, c$ are non-negative real numbers, and $\hat v_\alpha .\hat v_\beta = 0$, for $\alpha \ne \beta$ , then which of the following are true?
(A) $|[\hat v_1;\hat v_2;\hat v_3]| = 1$
(B) $a + b + c = 1$
(C) $\hat v_1 +\hat v_2 +\hat v_3 = 0$
(D) $\hat v_1;\hat v_2;\hat v_3$ are coplanar.
My approach: $\hat v_\alpha .\hat v_\beta = 0 \implies ab+bc+ca=0$ this cannot be as $a, b, c$ are non-negative real numbers.