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