Answer
$||2{\bf{e}} - 3{\bf{f}}|| = 4$.
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}}|| = \sqrt {\frac{3}{2}} $. So,
$||{\bf{e}} + {\bf{f}}|{|^2} = 1 + 2{\bf{e}}\cdot{\bf{f}} + 1 = \frac{3}{2}$
${\bf{e}}\cdot{\bf{f}} = - \frac{1}{4}$
Next, we evaluate $||2{\bf{e}} - 3{\bf{f}}|{|^2}$.
$||2{\bf{e}} - 3{\bf{f}}|{|^2} = \left( {2{\bf{e}} - 3{\bf{f}}} \right)\cdot\left( {2{\bf{e}} - 3{\bf{f}}} \right)$
$||2{\bf{e}} - 3{\bf{f}}|{|^2} = 4{\bf{e}}\cdot{\bf{e}} - 6{\bf{f}}\cdot{\bf{e}} - 6{\bf{e}}\cdot{\bf{f}} + 9{\bf{f}}\cdot{\bf{f}}$
$||2{\bf{e}} - 3{\bf{f}}|{|^2} = 4||{\bf{e}}|{|^2} - 12{\bf{e}}\cdot{\bf{f}} + 9||{\bf{f}}|{|^2}$
Substituting $||{\bf{e}}|| = 1$, $||{\bf{f}}|| = 1$ and ${\bf{e}}\cdot{\bf{f}} = - \frac{1}{4}$ in the last equation gives
$||2{\bf{e}} - 3{\bf{f}}|{|^2} = 4 - 12\left( { - \frac{1}{4}} \right) + 9 = 16$.
Thus, $||2{\bf{e}} - 3{\bf{f}}|| = 4$.
