Question

The vectors of magnitude $a,2a,3a$ meet at a point and their directions are along the diagonals of three adjacent faces of a cube. Then, the magnitude of their resultant is

• A. $5a$
• B. $6a$
• C. $10a$
• D. $9a$

Answer 1

The correct option is A

$5a$

Explanation for the correct option:

Consider the sides of the cube be along the axis, then diagonals have direction cosines as $\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0\right),\left(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}\right)$ and $\left(0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$

$\therefore$Resultant vector is $\frac{a}{\sqrt{2}}\left(\hat{i}+\hat{j}\right)+\frac{2a}{\sqrt{2}}\left(\hat{i}+\hat{j}\right)+\frac{3a}{2}\left(\hat{j}+\hat{k}\right)=\frac{a}{\sqrt{2}}\left(3\hat{i}+4\hat{j}+5\hat{k}\right)$

$\therefore$Magnitude of the resultant vector is $=\frac{a}{\sqrt{2}}\sqrt{9+16+25}$

$$\begin{gathered} =\frac{a}{\sqrt{2}}\sqrt{50} \\ =5a \end{gathered}$$

Hence the correct option is $\mathrm{Option}\left(\mathrm{A}\right)$