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

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a

$\frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{36}}$

b

$\hat{i} + \hat{j} + \hat{k}$

c

$\frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{2}}$

d

$\frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}}$

answer is D.

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Detailed Solution

Diagonal vector, $A = b \hat{i} + b \hat{j} + b \hat{k}$

$A = \sqrt{b^{2} + b^{2} + b^{2}} = \sqrt{3} b$

$\hat{A} = \frac{A}{A} = \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}}$

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