Question

Let us define the length of a vector $\widehat{ai}+\widehat{bj}+\widehat{ck}$ as $|a|+|b|+|c|.$ This definition coincides with the usual definition of length of a vector $\widehat{ai}+\widehat{bj}+\widehat{ck}$ if and only if

• A. a=b=c=0
• B. any two of a,b and c are zero
• C. any one of a,b and c is zero
• D. a+b+c=0

Answer 1

The correct option is A a=b=c=0

We have,

Let the length of the vector $\overrightarrow{r}=a\hat{i}+b\hat{j}+c\hat{k}$.

Then,

$\left|\overrightarrow{r}\right|=\sqrt{a^2+b^2+c^2}$

We can write this,

$\left|\overrightarrow{r}\right|=\sqrt{a^2+b^2+c^2}\approx \left|a\right|+\left|b\right|+\left|c\right|$

If

$\overrightarrow{c}=0,then$

$\left|\overrightarrow{r}\right|=\sqrt{a^2+b^2+0}\approx \left|a\right|+\left|b\right|+0$

$\sqrt{a^2+b^2}=\left|a\right|+\left|b\right|$

\end{align}\]

If

\[\begin{align}

$a=b=0,then$

$\left|\overrightarrow{r}\right|=\sqrt{0+0+c^2}\approx 0+0+c$

$\sqrt{c^2}=\left|c\right|$

Hence, this is the aswer.