Attempt 1: 10 attempts remaining. Find the value of \( k \) for which the vectors \( \left[\begin{array}{c}-1 \\ -2 \\ -4 \\ -3\end{array}\right] \) and \( \left[\begin{array}{c}4 \\ -5 \\ -4 \\ k\end{array}\right] \) are orthogonal. \( k= \)

To find the value of \( k \) for which the vectors \( \left[\begin{array}{c}-1 \\ -2 \\ -4 \\ -3\end{array}\right] \) and \( \left[\begin{array}{c}4 \\ -5 \\ -4 \\ k\end{array}\right] \) are orthogonal, we need to set their dot product to zero. The dot product is calculated as follows: \[ (-1) \cdot 4 + (-2) \cdot (-5) + (-4) \cdot (-4) + (-3) \cdot k = 0. \] Calculating each part: \[ -4 + 10 + 16 - 3k = 0. \] Combining the constant terms: \[ 22 - 3k = 0. \] Now, solving for \( k \): \[ 3k = 22 \implies k = \frac{22}{3}. \] Thus, the value of \( k \) for which the vectors are orthogonal is \[ k = \frac{22}{3}. \]