Question

Let us define the length of a vector $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ as $\left|\alpha \right|+\left|\beta \right|+\left|\gamma \right|$. This definition coincides with the usual definition of length of vector $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ if and only if

• A. Any two of $\alpha$, $\beta$, $\gamma$ are zero
• B. Any one of $\alpha$, $\beta$, $\gamma$ are zero
• C. $a=b=c=0$
• D. $a+b+c=0$

Answer 1

Answer: (A)

Any two of $\alpha$, $\beta$, $\gamma$ are zero

According to original definition
$|\alpha i+\beta j+\gamma k|$ $=\sqrt{{\alpha }^2+{\beta }^2+{\gamma }^2}$
$=|\alpha |+|\beta |+|\gamma |$.
Squaring both sides give us
${\alpha }^2+{\beta }^2+{\gamma }^2=(|\alpha |+|\beta |+|\gamma |{)}^2$
$={\alpha }^2+{\beta }^2+{\gamma }^2+2(|\alpha ||\beta |+|\beta ||\gamma |+|\gamma ||\alpha |)$
Hence, ${\alpha }^2+{\beta }^2+{\gamma }^2={\alpha }^2+{\beta }^2+{\gamma }^2+2(|\alpha ||\beta |+|\beta ||\gamma |+|\gamma ||\alpha |)$
$(|\alpha ||\beta |+|\beta ||\gamma |+|\gamma ||\alpha |)=0$
But $|\alpha ||\beta |+|\beta ||\gamma |+|\gamma ||\alpha |\ge 0$.
Hence if any two of $\alpha ,\beta ,\gamma$ is $0$, then
$|\alpha ||\beta |+|\beta ||\gamma |+|\gamma ||\alpha |=0$
$|\alpha i+\beta j+\gamma k|=|\alpha |+|\beta |+|\gamma |$