1. **Problem 1: Find $6 \overline{A} + \overline{B}$** Given vectors: $$\overline{A} = 4i + 3j - k$$ $$\overline{B} = -2i - 5j + 3k$$ Step 1: Multiply vector $\overline{A}$ by 6: $$6 \overline{A} = 6(4i + 3j - k) = 24i + 18j - 6k$$ Step 2: Add vector $\overline{B}$: $$6 \overline{A} + \overline{B} = (24i + 18j - 6k) + (-2i - 5j + 3k)$$ Step 3: Combine like terms: $$= (24 - 2)i + (18 - 5)j + (-6 + 3)k = 22i + 13j - 3k$$ 2. **Problem 2: Find $-3 \overline{B} - 5 \overline{A}$** Step 1: Multiply vector $\overline{B}$ by -3: $$-3 \overline{B} = -3(-2i - 5j + 3k) = 6i + 15j - 9k$$ Step 2: Multiply vector $\overline{A}$ by -5: $$-5 \overline{A} = -5(4i + 3j - k) = -20i - 15j + 5k$$ Step 3: Add the two results: $$-3 \overline{B} - 5 \overline{A} = (6i + 15j - 9k) + (-20i - 15j + 5k)$$ Step 4: Combine like terms: $$= (6 - 20)i + (15 - 15)j + (-9 + 5)k = -14i + 0j - 4k = -14i - 4k$$ **Final answers:** 1. $6 \overline{A} + \overline{B} = 22i + 13j - 3k$ 2. $-3 \overline{B} - 5 \overline{A} = -14i - 4k$