Answer
Substituting $||{\bf{e}}|| = 1$, $||{\bf{f}}|| = 1$ and ${\bf{e}}\cdot{\bf{f}} = \frac{1}{8}$ in
$||{\bf{e}} - {\bf{f}}|{|^2} = ||{\bf{e}}|{|^2} - 2{\bf{e}}\cdot{\bf{f}} + ||{\bf{f}}|{|^2}$
gives
$||{\bf{e}} - {\bf{f}}|| = \frac{{\sqrt 7 }}{2}$
Work Step by Step
We have
$||{\bf{e}} + {\bf{f}}|{|^2} = \left( {{\bf{e}} + {\bf{f}}} \right)\cdot\left( {{\bf{e}} + {\bf{f}}} \right) = {\bf{e}}\cdot{\bf{e}} + {\bf{f}}\cdot{\bf{e}} + {\bf{e}}\cdot{\bf{f}} + {\bf{f}}\cdot{\bf{f}}$
$||{\bf{e}} + {\bf{f}}|{|^2} = ||{\bf{e}}|{|^2} + 2{\bf{e}}\cdot{\bf{f}} + ||{\bf{f}}|{|^2}$
Since ${\bf{e}}$ and ${\bf{f}}$ are unit vectors, $||{\bf{e}}|| = 1$ and $||{\bf{f}}|| = 1$, and we have $||{\bf{e}} + {\bf{f}}|| = \frac{3}{2}$. So,
$||{\bf{e}} + {\bf{f}}|{|^2} = 1 + 2{\bf{e}}\cdot{\bf{f}} + 1 = \frac{9}{4}$
${\bf{e}}\cdot{\bf{f}} = \frac{1}{8}$
Next, we evaluate $||{\bf{e}} - {\bf{f}}|{|^2}$.
$||{\bf{e}} - {\bf{f}}|{|^2} = \left( {{\bf{e}} - {\bf{f}}} \right)\cdot\left( {{\bf{e}} - {\bf{f}}} \right) = {\bf{e}}\cdot{\bf{e}} - {\bf{f}}\cdot{\bf{e}} - {\bf{e}}\cdot{\bf{f}} + {\bf{f}}\cdot{\bf{f}}$
$||{\bf{e}} - {\bf{f}}|{|^2} = ||{\bf{e}}|{|^2} - 2{\bf{e}}\cdot{\bf{f}} + ||{\bf{f}}|{|^2}$
Substituting $||{\bf{e}}|| = 1$, $||{\bf{f}}|| = 1$ and ${\bf{e}}\cdot{\bf{f}} = \frac{1}{8}$ in the last equation gives
$||{\bf{e}} - {\bf{f}}|{|^2} = 1 - \frac{1}{4} + 1 = \frac{7}{4}$.
Thus, $||{\bf{e}} - {\bf{f}}|| = \frac{{\sqrt 7 }}{2}$.
