Chapter 9.2, Problem 40E

To determine

To prove: The given property.

Explanation of Solution

Given information:

The property is $\left(au\right)\cdot v=a\left(u\cdot v\right)=u\cdot \left(av\right)$ .

Calculation:

Consider the vector.

  $\begin{array}{l}u=\langle x_1,y_1\rangle \\ v=\langle x_2,y_2\rangle \end{array}$

The value of $\left(au\right)\cdot v$ is,

  $\left(au\right)\cdot v=a{x}_1{x}_2+a{y}_1{y}_2$

The value of $a\left(u\cdot v\right)$ is,

  $a\left(u\cdot v\right)=a{x}_2{x}_1+a{y}_2{y}_1$

The value of $u\cdot \left(av\right)$ is,

  $u\cdot \left(av\right)=a{x}_2{x}_1+a{y}_2{y}_1$

Thus, $\left(au\right)\cdot v=a\left(u\cdot v\right)=u\cdot \left(av\right)$ .