Question

To determine the meaning of the dot product \({\rm{A}} \cdot {\rm{P}}\).

Short answer

The meaning of the dot product \({\rm{A}} \cdot {\rm{P}}\) is the vendor's total earn per day.

Step-by-step solution

Concept of Dot Product

Formula used:

Consider a general expression to find the dot product between three-dimensional vectors.

\(\begin{aligned}{l}{\rm{A}} \cdot {\rm{P}} &= \langle a,b,c\rangle \cdot \left\langle {{p_1},{p_2},{p_3}} \right\rangle \\{\rm{A}} \cdot {\rm{P}} &= a{p_1} + b{p_2} + c{p_3}\end{aligned}\)

Calculation for the meaning of the dot product\({\rm{A}} \cdot {\rm{P}}\)

The minimum of two vectors are required to perform a dot product. The resultant dot product of two vectors is scalar. hence, the dot product is also known as a scalar product.

The number of hamburgers, hot dogs, and soft drinks sold by street vendor are \(a,{\rm{ }}b,{\rm{ }}c\)respectively. The vector form of the items sold by vendor is denoted by \({\rm{A}} = \langle a,b,c\rangle \).

The cost of hamburgers, hot dogs, and soft drinks are \(2,1.50\), and \(1\)respectively and its vector form is denoted by \({\rm{P}} = \langle 2,1.5,1\rangle \).

Substitute the value of components \({p_1} = 2,{p_2} = 1.5\), and \({p_3} = 1\) in the above result.

\(\begin{aligned}{l}{\rm{A}} \cdot {\rm{P}} &= \langle a,b,c\rangle \cdot \langle 2,1.5,1\rangle \\{\rm{A}} \cdot {\rm{P}} &= a(2) + b(1.5) + c(1)\\{\rm{A}} \cdot {\rm{P}} &= \left( {\begin{aligned}{*{20}{c}}{({\rm{ Numberof hamburgers sold }})({\rm{ price of each hamburger }}) + }\\{({\rm{ Numberof hotdogs sold }})({\rm{ priceof each hotdog }}) + }\\{({\rm{ Numberof softdrinks sold }})({\rm{ price of eachsoftdrink }})}\\{ = {\rm{ Vendorstotalearnperday }}}\end{aligned}} \right)\end{aligned}\)

Thus, the meaning of the dot product \({\rm{A}} \cdot {\rm{P}}\) is the vendor's total earn per day.