Let us define the length of a vector $a\hat i +b\hat j+c\hat k$ as $|a|+|b|+|c|$. This definition coincides with the usual definition of length of a vector $a\hat i +b\hat j+c\hat k$ iff

any two of a, b and c are zero

We have,

$|a\hat i +b\hat j + c\hat k|=|a|+|b|+|c|$

$⇒\sqrt{a^2+b^2+c^2}=|a|+|b|+|c|$

$⇒a^2+b^2+c^2=a^2+b^2+c^2+2(|a||b|+|b||c|+|c||a|)$

$⇒|a||b|+|b||c|+|c||a|=0$

$⇒ab=bc = ca=0$

⇒ Any two of a, b, c are zero.