8. จงหาเวกเตอร์ที่ตั้งฉากทั้ง \( \overline{\mathrm{u}} \) และ \( \overline{\mathrm{v}} \) 1) \( \overline{\mathrm{u}}=-3 \overline{\mathrm{i}}+2 \overline{\mathrm{j}}-2 \overline{\mathrm{k}} \) และ \( \overline{\mathrm{v}}=\overline{\mathrm{i}}-4 \overline{\mathrm{j}}+3 \overline{\mathrm{k}} \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) 2) \( \overline{\mathbf{u}}=\overline{\mathrm{i}}+\overline{\mathrm{j}}-\overline{\mathrm{k}} \) และ \( \overline{\mathbf{v}}=\overline{\mathrm{j}} \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \)

Alright, I need to find vectors that are perpendicular to both vectors **u** and **v**. Let's tackle the first problem first. **Problem 1:** Given: - **u** = -3i + 2j - 2k - **v** = i - 4j + 3k I need to find a vector **w** such that **w** is perpendicular to both **u** and **v**. To find such a vector, I can use the cross product of **u** and **v**. The cross product of two vectors results in a vector that is perpendicular to both of them. So, let's compute **u** × **v**. **u** = (-3, 2, -2) **v** = (1, -4, 3) The cross product **u** × **v** is calculated as follows: |i j k| |-3 2 -2| |1 -4 3| Calculating the determinant: i-component: (2*3) - (-2*-4) = 6 - 8 = -2 j-component: -[( -3*3 ) - ( -2*1 )] = -[-9 - (-2)] = -[-7] = 7 k-component: [(-3*-4) - (2*1)] = 12 - 2 = 10 So, **u** × **v** = -2i + 7j + 10k Therefore, the vector **w** that is perpendicular to both **u** and **v** is **w** = -2i + 7j + 10k. **Problem 2:** Given: - **u** = i + j - k - **v** = j I need to find a vector **w** such that **w** is perpendicular to both **u** and **v**. First, let's express **v** in terms of i, j, and k: **v** = 0i + 1j + 0k Now, compute **u** × **v**. **u** = (1, 1, -1) **v** = (0, 1, 0) |i j k| |1 1 -1| |0 1 0| Calculating the determinant: i-component: (1*0) - (-1*1) = 0 + 1 = 1 j-component: -[(1*0) - (-1*0)] = -[0 - 0] = 0 k-component: [(1*1) - (1*0)] = 1 - 0 = 1 So, **u** × **v** = 1i + 0j + 1k Therefore, the vector **w** that is perpendicular to both **u** and **v** is **w** = i + 0j + k, which simplifies to **w** = i + k.