1. **State the problem:** Find the value of $k$ such that vectors $U = (2, 3k, -4, 1, 5)$ and $V = (6, -1, 3, 7, 2k)$ are orthogonal. 2. **Recall the definition of orthogonal vectors:** Two vectors are orthogonal if their dot product is zero. 3. **Compute the dot product:** $$U \cdot V = 2 \times 6 + (3k) \times (-1) + (-4) \times 3 + 1 \times 7 + 5 \times (2k)$$ 4. **Simplify the expression:** $$= 12 - 3k - 12 + 7 + 10k$$ 5. **Combine like terms:** $$= (12 - 12 + 7) + (-3k + 10k) = 7 + 7k$$ 6. **Set the dot product equal to zero for orthogonality:** $$7 + 7k = 0$$ 7. **Solve for $k$:** $$7k = -7$$ $$k = -1$$ **Final answer:** $k = -1$