# A vector $a\hat i+b\hat j+c\hat k$ is such that its magnitude is $|a|+|b|+|c|$. This is possible only if

$(a)\:a=b=c=0\:\:\qquad\:\:(b)\:any\:two\:of\:a,b\:and\:c\:are\:zero\:\:\qquad\:\:(c)\:any\:one\:of\:a,b,c\:is\:zero\:\:\qquad\:\:(d)\:a+b+c=0$

Given $|\overrightarrow a\hat i+b\hat j+c\hat k|=|a|+|b|+|c|$
$\Rightarrow\:a^2+b^2+c^2=a^2+b^2+c^2+2(|a|.|b|+|b|.|c|+|c|.|a|)$
$\Rightarrow\:|a|.|b|+|b|.|c|+|c|.|a|=0$
This is possible only if $ab=bc=ca=0$
$\Rightarrow\:$ Any two of $a,b\:and\:c$ should be zero.