Question

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

• A. $3a$
• B. $5a$
• C. $2a$
• D. $4a$

Answer 1

The correct option is B $5a$

Solution:

Let the vectors of magnitudes $a,2a,3a$ act along $OP,OQ,OR$ respectively.Then vectors are $OP,OQ,OR$ are

$a\left(\frac{\overrightarrow{i}+\overrightarrow{j}}{\sqrt{2}}\right),$$2a\left(\frac{\overrightarrow{j}+\overrightarrow{k}}{\sqrt{2}}\right),$$3a\left(\frac{\overrightarrow{k}+\overrightarrow{i}}{\sqrt{2}}\right)$ respectively.

Their resultant say $R$ is given by

$\overrightarrow{R}=$$a\left(\frac{\overrightarrow{i}+\overrightarrow{j}}{\sqrt{2}}\right)+$$2a\left(\frac{\overrightarrow{j}+\overrightarrow{k}}{\sqrt{2}}\right)+$$3a\left(\frac{\overrightarrow{k}+\overrightarrow{i}}{\sqrt{2}}\right)$

$=\frac{a}{\sqrt{2}}(4\overrightarrow{i}+3\overrightarrow{j}+5\overrightarrow{k})$

$\therefore |\overrightarrow{R}|=\sqrt{\frac{a^2}{2}(16+9+25)}=5a$

Hence, B is the correct answer.