In what ratio, the spacings of three planes 100, 110, and 111 of a cubic lattice vary with each other? 

Formula used:

${\mathrm{d}}_{\mathrm{hkl}}=\frac{\mathrm{a}}{{\left({\mathrm{h}}^2+{\mathrm{k}}^2+{\mathrm{l}}^2\right)}^{1/2}}$

For 110 plane:

  $$\begin{gathered} d_{100}=\frac{a}{{\left(1^2+0^2+0^2\right)}^{1/2}} \\ =\mathrm{a} \end{gathered}$$ 

For 110 plane:

 $$\begin{gathered} d_{110}=\frac{a}{{\left(1^2+1^2+0^2\right)}^{1/2}} \\ =\frac{a}{\sqrt{2}} \end{gathered}$$                           

For 111 Plane:

$$\begin{gathered} d_{111}=\frac{a}{{\left(1^2+1^2+1^2\right)}^{1/2}} \\ =\frac{a}{\sqrt{3}} \end{gathered}$$ 

Ratio becomes:

$\begin{array}{ccc}{d}_{100}:d_{110}:d_{111}& =& a:\frac{a}{\sqrt{2}}:\frac{a}{\sqrt{3}}\\ & =& 1:0.707:0.577\end{array}$