Let \( \vec{a}=\langle-2,5,-1\rangle \) and \( \vec{b}=\langle-1,-2, k\rangle \). Find \( k \) so that \( \vec{a} \) and \( \vec{b} \) will be orthogonal (form a 90 degree angle). \( k= \)

To find the value of \( k \) that makes the vectors \( \vec{a} \) and \( \vec{b} \) orthogonal, we need to set their dot product to zero. The dot product of \( \vec{a} \) and \( \vec{b} \) is computed as follows: \[ \vec{a} \cdot \vec{b} = (-2)(-1) + (5)(-2) + (-1)(k) = 2 - 10 - k \] Setting the dot product equal to zero for orthogonality, we have: \[ 2 - 10 - k = 0 \] Simplifying this gives: \[ -8 - k = 0 \implies k = -8 \] Thus, the value of \( k \) is \( \boxed{-8} \).