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
$a \hat{i} + b \hat{j} + c \hat{k}$ if and only if

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a=b=c=0

any two of a, b and c are zero

any one of a, b and c is zero

a+b+c=0

answer is B.

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Detailed Solution

$\begin{array}{l} | a | + | b | + | c | = \sqrt{a^{2} + b^{2} + c^{2}} \\ \Leftrightarrow 2 | ab | + 2 | bc | + 2 | ca | = 0 \\ \Leftrightarrow ab = bc = ca = 0 \Leftrightarrow any two of a, b and c are zero \end{array}$

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