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Question

If three vectors along coordinate axis represents the adjacent sides of a cube of length b, then the unit vector along its diagonal passing through the origin will be


A
^i+^j+^k2
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B
^i+^j+^k36
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C
^i+^j+^k
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D
^i+^j+^k3
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Solution

The correct option is C $$ \frac { \hat { i } +\hat { j } +\hat { k } }{ \sqrt { 3 } } $$

Point diagonal line$$(0,0,0)$$  and $$(b,b,b)$$

Vector along diagonal, $$\vec{B}=(b-o)\hat{i}+(b-o)\hat{j}+(b-o)\hat{k}=b\hat{i}+b\hat{j}+b\hat{k}$$

Unit vector along diagonal, $$\hat{B}=\dfrac{{\vec{B}}}{\left| {\vec{B}} \right|}=\dfrac{b\hat{i}+b\hat{j}+b\hat{k}}{\sqrt{{{b}^{2}}+{{b}^{2}}+{{b}^{2}}}}=\dfrac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$$

Hence, unit vector along diagonal is $$\dfrac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$$


Physics

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