Find the value of $k$ for which the vectors $\left[\begin{array}{c}4 \\ 4 \\ 3 \\ -5\end{array}\right]$ and $\left[\begin{array}{c}1 \\ -2 \\ -2 \\ k\end{array}\right]$ are orthogonal.
\[
k=\square
\]

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 \(k = \boxed{-2}\).