Property of the dot product?

Question

If I have vectors $v_1, v_2$ and scalar, real $a$

How is :

$$\overrightarrow{v_1} \cdot a\overrightarrow{v_{2}} = a\overrightarrow{v_1} \cdot \overrightarrow{v_{2}}$$

I don't see how this property holds?

Answer 1

Suppose $\overrightarrow{v_{1}} = x_{1} \hat{\textbf{i}} + y_{1} \hat{\textbf{j}} + z_{1} \hat{\textbf{k}} $ and $\overrightarrow{v_{2}} = x_{2} \hat{\textbf{i}} + y_{2} \hat{\textbf{j}} + z_{2} \hat{\textbf{k}} $ , then consider the L.H.S \ $\overrightarrow{v_{1}}.a\overrightarrow{v_{2}} = (x_{1} \hat{\textbf{i}} + y_{1} \hat{\textbf{j}} + z_{1} \hat{\textbf{k}}).(ax_{2} \hat{\textbf{i}} + ay_{2} \hat{\textbf{j}} + az_{2} \hat{\textbf{k}}) = x_{1}ax_{2} + y_{1}ay_{2}+z_{1}az_{2}$ which is the same when we consider the right hand side , because here in the result the multiplication is commutative as it is in real system.