Step 1 :Two vectors are orthogonal if their dot product is zero. The dot product of two vectors is calculated by multiplying corresponding entries and then summing those products.
Step 2 :So, we need to find the value of \(k\) such that the dot product of the two given vectors is zero.
Step 3 :The vectors are \(\left[\begin{array}{c}4 \ 4 \ 3 \ -5\end{array}\right]\) and \(\left[\begin{array}{c}1 \ -2 \ -2 \ k\end{array}\right]\).
Step 4 :The dot product of these vectors is \(-5*k - 10\).
Step 5 :We solve the equation \(-5*k - 10 = 0\) for \(k\).
Step 6 :The solution to this equation is \(k = -2\).
Step 7 :Final Answer: The value of \(k\) for which the vectors are orthogonal is \(k = \boxed{-2}\).