$1.[4$ points$]$ In a multiple channel marketing campaign, potential customers are divided into $\backslash ($ m $\backslash )$ market segments, which are groups of customers with similar demographics; for example, male, female, married, college educated women aged $\backslash (25-29\backslash )$ etc. A company markets its products by purchasing advertising in a set of $\backslash ($ n $\backslash )$ channels; for example, specific TV or radio shows, magazines, web sites, blogs, direct mail, and so on$.$ The ability of each channel to deliver impressions or views by potential customers is characterized by the data matrix $\backslash (\backslash$mathbf$\{$X$\}\backslash ),$ which has $\backslash ($ m $\backslash )$ rows and $\backslash ($ n $\backslash )$ columns representing $\backslash ($ m $\backslash )$ customer segments and $\backslash ($ n $\backslash )$ advertising channels. The entry in the $\backslash ($ i $\backslash )$ th row and $\backslash ($ j $\backslash )$ th column of the data matrix is the number of views of customers in segment $\backslash ($ i $\backslash )$ for each Rupee spent on channel $\backslash ($ j $\backslash ).$ The $\backslash ($ n $\backslash )-$vector $\backslash (\backslash$mathbf$\{$c$\}\backslash )$ will denote the company's purchases of advertising, in Rupees, in the $\backslash ($ n $\backslash )$ channels. Finally, we introduce the $\backslash ($ m $\backslash )-$vector a$,$ where $\backslash ($ a$\_\{$i$\}\backslash )$ gives the profit in Rupees per impression in market segment $\backslash ($ i $\backslash ).$

$($a$)$ Express the total amount of money the company spends on advertising as a dot product of two vectors. Explain your answer briefly.

$($b$)$ Interpret the quantity Xc in plain English and using the data as reference briefly; do this by starting with interpreting the underhighlighted terms below in plain English:

$($c$)$ The total profit of the company is the dot product of two vectors. What are they? Explain your answer briefly.