Question

Let us define, the length of a vector $a\overline{i}+b\overline{j}+c\overline{k}$ as $|a|+|b|+|c|$. This definition coincides with the usual definition of the length of a vector $a\overline{i}+b\overline{j}+c\overline{k}$ if

• A. $a=b=c=0$
• B. Any one of $a,b,c$, is zero
• C. Any two of $a,b,c$ are zero
• D. $a=b=c\ne 0$

Answer 1

The correct option is C Any two of $a,b,c$ are zero

The definitions will coincides if $|a|+|b|+|c|=\sqrt{a^2+b^2+c^2}$

$\Rightarrow a^2+b^2+c^2+2(|a||b|+|b||c|+|c||a|)=a^2+b^2+c^2$, square both sides

$\Rightarrow |a||b|+|b||c|+|c||a|=0$

Clearly above equation will satisfy if any two of $a,b,c$ are zero.