Three vectors of magnitudes $a, 2 a, 3 a$ are along the directions of the diagonals of 3 adjacent faces of a…
Three vectors of magnitudes $a, 2 a, 3 a$ are along the directions of the diagonals of 3 adjacent faces of a cube that meet in a point. Then, the magnitude of the sum of those diagonals is
$4 a$
$5 a$
$6 a$
$8 a$
Solution
Let the vectors of magnitude $a, 2 a, 3 a$ are along $O P, O Q, O R$, respectively.
Then, vectors are $O P, O Q, O R$ are
$a\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right), 2 a\left(\frac{\hat{j}+\hat{k}}{\sqrt{2}}\right), 3 a\left(\frac{\hat{k}+\hat{i}}{\sqrt{2}}\right)$ respectively.
Their resultant say $R$ is given by
$\mathbf{R}=a\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)+2 a\left(\frac{\hat{j}+\hat{k}}{\sqrt{2}}\right)+3 a\left(\frac{\hat{k}+\hat{i}}{\sqrt{2}}\right)$
$=\frac{a}{\sqrt{2}}(4 \hat{i}+3 \hat{j}+5 \hat{k})$
$\therefore|\mathbf{R}|=\sqrt{\frac{a^2}{2}(16+9+25)}=5 a$