1. **Problem Statement:** Determine which pairs of vectors from the given list are orthogonal. Two vectors are orthogonal if their dot product equals zero. 2. **Formula:** The dot product of vectors $\mathbf{a} = [a_1, a_2, a_3]$ and $\mathbf{b} = [b_1, b_2, b_3]$ is given by: $$\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$$ If $\mathbf{a} \cdot \mathbf{b} = 0$, then $\mathbf{a}$ and $\mathbf{b}$ are orthogonal. 3. **Calculate dot products for each pair:** - Pair 1: $[4, -2, 2]$ and $[3, 15, 13]$ $$4 \times 3 + (-2) \times 15 + 2 \times 13 = 12 - 30 + 26 = 8$$ Not zero, so not orthogonal. - Pair 2: $[1, -5, 3]$ and $[-17, 8, 19]$ $$1 \times (-17) + (-5) \times 8 + 3 \times 19 = -17 - 40 + 57 = 0$$ Dot product is zero, so orthogonal. - Pair 3: $[3, 1, 2]$ and $[8, -8, -8]$ $$3 \times 8 + 1 \times (-8) + 2 \times (-8) = 24 - 8 - 16 = 0$$ Dot product is zero, so orthogonal. - Pair 4: $[1, -5, 2]$ and $[6, -8, -23]$ $$1 \times 6 + (-5) \times (-8) + 2 \times (-23) = 6 + 40 - 46 = 0$$ Dot product is zero, so orthogonal. - Pair 5: $[-5, -5, -1]$ and $[-10, 11, -2]$ $$-5 \times (-10) + (-5) \times 11 + (-1) \times (-2) = 50 - 55 + 2 = -3$$ Not zero, so not orthogonal. 4. **Answer:** The orthogonal pairs are: - $[1, -5, 3]$ and $[-17, 8, 19]$ - $[3, 1, 2]$ and $[8, -8, -8]$ - $[1, -5, 2]$ and $[6, -8, -23]$