1. **Problem Statement:** Given two vectors $u$ and $v$, find the value of $k$ such that their dot product $u \cdot v = 0$, meaning $u$ and $v$ are orthogonal. 2. **Vectors:** $$u = (1, 2, 3, 4), \quad v = (6, k, -8, 2)$$ 3. **Dot Product Formula:** The dot product of two vectors $u = (u_1, u_2, u_3, u_4)$ and $v = (v_1, v_2, v_3, v_4)$ is $$u \cdot v = u_1 v_1 + u_2 v_2 + u_3 v_3 + u_4 v_4$$ 4. **Calculate the dot product:** Substitute the components: $$u \cdot v = 1 \times 6 + 2 \times k + 3 \times (-8) + 4 \times 2$$ Simplify: $$= 6 + 2k - 24 + 8$$ $$= (6 - 24 + 8) + 2k = (-10) + 2k$$ 5. **Set dot product to zero for orthogonality:** $$-10 + 2k = 0$$ 6. **Solve for $k$:** Add 10 to both sides: $$2k = 10$$ Divide both sides by 2: $$k = 5$$ **Final answer:** $$\boxed{5}$$ This means when $k=5$, vectors $u$ and $v$ are orthogonal.