Answer
${\bf{u}}\cdot\left( {{\bf{v}} + {\bf{w}}} \right) = {\bf{u}}\cdot{\bf{v}} + {\bf{u}}\cdot{\bf{w}}$
Work Step by Step
Let the components of ${\bf{u}}$, ${\bf{v}}$ and ${\bf{w}}$ be given by,
${\bf{u}} = \left( {{u_1},{u_2},{u_3}} \right)$, ${\ }$ ${\bf{v}} = \left( {{v_1},{v_2},{v_3}} \right)$, ${\ }$ ${\bf{w}} = \left( {{w_1},{w_2},{w_3}} \right)$.
So,
${\bf{u}}\cdot\left( {{\bf{v}} + {\bf{w}}} \right) = \left( {{u_1},{u_2},{u_3}} \right)\cdot\left( {{v_1} + {w_1},{v_2} + {w_2},{v_3} + {w_3}} \right)$
${\bf{u}}\cdot\left( {{\bf{v}} + {\bf{w}}} \right) = {u_1}\left( {{v_1} + {w_1}} \right) + {u_2}\left( {{v_2} + {w_2}} \right) + {u_3}\left( {{v_3} + {w_3}} \right)$
${\bf{u}}\cdot\left( {{\bf{v}} + {\bf{w}}} \right) = {u_1}{v_1} + {u_1}{w_1} + {u_2}{v_2} + {u_2}{w_2} + {u_3}{v_3} + {u_3}{w_3}$
${\bf{u}}\cdot\left( {{\bf{v}} + {\bf{w}}} \right) = {u_1}{v_1} + {u_2}{v_2} + {u_3}{v_3} + {u_1}{w_1} + {u_2}{w_2} + {u_3}{w_3}$
Since ${\bf{u}}\cdot{\bf{v}} = {u_1}{v_1} + {u_2}{v_2} + {u_3}{v_3}$ and ${\bf{u}}\cdot{\bf{w}} = {u_1}{w_1} + {u_2}{w_2} + {u_3}{w_3}$, therefore
${\bf{u}}\cdot\left( {{\bf{v}} + {\bf{w}}} \right) = {\bf{u}}\cdot{\bf{v}} + {\bf{u}}\cdot{\bf{w}}$
