Chapter 6.4, Problem 72E

a.

To determine

To calculate:

Dot product of $u.v$ and its mening.

Answer to Problem 72E

Dot product of $u.v$ is $11877.5$ that means numbers of hats and sweatshirts is $11877.5$

Explanation of Solution

Given information:

  $\begin{array}{l}u=\langle 8,13\rangle \\ v=\langle 13.25,20.50\rangle \end{array}$

  $u.v$

Calculation:

Vector $u=\langle 8,13\rangle$ gives the numbers of hats and sweatshirts,

To find the value of $u.v$ ,

By using rule,

  $\begin{array}{l}u.v=\langle 3140,2750\rangle .\langle 2.25,1.75\rangle \\ =3140\left(2.25\right)+2750\left(1.75\right)\\ =7065+4812.5\\ =11877.5\end{array}$

So, numbers of hats and sweatshirts is $11877.5$

Hence the dot product of $u.v$ is where $u=\langle 8,13\rangle ,v=\langle 13.25,20.50\rangle$ is $11877.5$ .

b.

To determine

To identify:

Vector operation used to decrease prices by $0.5\%$

Answer to Problem 72E

Scalar multiplication is vector operation which is used to decrease prices by $0.5\%$ .

Explanation of Solution

Given information:

  $\begin{array}{l}u=\langle 8,13\rangle \\ v=\langle 13.25,20.50\rangle \end{array}$

  $u.v$

Calculation:

By using rule of scalar multiplication,

  $\begin{array}{l}c\left(u.v\right)=cu.v\\ =u.cv\end{array}$

Therefore, to decrease the prices by $0.5\%$ .

Vector operation used is scalar multiplication.

It gives new total cost of number of sweat shirts.

  $\begin{array}{l}c\left(u.v\right)=cu.v\\ u\left(1.005\right)v=\left(1.005\right)\left(u.v\right)\end{array}$

Hence, Vector operation used to decrease prices by $0.5\%$ is scalar multiplication.