Question

The projections of a line segment on $x,yandz$ axes are respectively $\sqrt{2},3,5$. The length of the line segment is

• A. $6$
• B. $11$
• C. $8$
• D. $5$

Answer 1

The correct option is A $6$

Let the line segment be represented in vector form as $\overrightarrow{a}=a_1\overrightarrow{i}+a_2\overrightarrow{j}+a_3\overrightarrow{k}$

Vector along coordinate axes are $\overrightarrow{i},\overrightarrow{j},\overrightarrow{k}$

Given that projection of $\overrightarrow{a}$ on x axis is $\sqrt{3}$, that with y axis is $3$ and that with z axis is $5$.

$⟹\overrightarrow{a}.\overrightarrow{i}$ = $\sqrt{2}$, $\therefore a_1=\sqrt{2}$

similarily, $a_2=3,a_3=5$

$\therefore \overrightarrow{a}=\sqrt{2}\overrightarrow{i}+3\overrightarrow{j}+5\overrightarrow{k}$

the length of the line segment=$|\overrightarrow{a}|=\sqrt{2+9+25}=\sqrt{3}6=6$