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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
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B
5a
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C
2a
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D
4a
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Solution

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 
$$a\left(\cfrac{\vec i+\vec j}{\sqrt2}\right),$$$$2a\left(\cfrac{\vec j+\vec k}{\sqrt2}\right),$$$$3a\left(\cfrac{\vec k+\vec i}{\sqrt2}\right)$$ respectively.
Their resultant say $$R$$ is given by
$$\vec R =$$$$a\left(\cfrac{\vec i+\vec j}{\sqrt2}\right)+$$$$2a\left(\cfrac{\vec j+\vec k}{\sqrt2}\right)+$$$$3a\left(\cfrac{\vec k+\vec i}{\sqrt2}\right)$$
$$=\cfrac{a}{\sqrt2}(4\vec i+3\vec j+5\vec k)$$
$$\therefore |\vec R|=\sqrt{\cfrac{a^2}2(16+9+25)}=5a$$
Hence, B is the correct answer.



642942_37676_ans_20ecbf9af7564ee89b941d5ad7750685.png

Mathematics

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