1. **Problem Statement:** Write each point A, B, C, D as a sum of scalar multiples of vectors $v_1$ and $v_2$. 2. **Formula and Rules:** Each point in the plane can be expressed as $\mathbf{p} = a v_1 + b v_2$ where $a,b$ are scalars. 3. **Given:** $A = 4v_1 + 3v_2$ $B = -3v_1$ $C = -2v_1 - 1v_2$ $D = 3v_1 + v_2$ 4. **Vectors between points:** Vector $AB = B - A = (-3v_1) - (4v_1 + 3v_2) = -7v_1 - 3v_2$ 5. **Calculate $BA - CA$:** $BA = A - B = (4v_1 + 3v_2) - (-3v_1) = 7v_1 + 3v_2$ $CA = A - C = (4v_1 + 3v_2) - (-2v_1 - 1v_2) = 6v_1 + 4v_2$ So, $$BA - CA = (7v_1 + 3v_2) - (6v_1 + 4v_2) = (7 - 6)v_1 + (3 - 4)v_2 = v_1 - v_2$$ 6. **Calculate $C + 7v_2$:** $$C + 7v_2 = (-2v_1 - v_2) + 7v_2 = -2v_1 + 6v_2$$ 7. **Calculate $D + 7DA$:** First find $DA = A - D = (4v_1 + 3v_2) - (3v_1 + v_2) = v_1 + 2v_2$ Then, $$7DA = 7(v_1 + 2v_2) = 7v_1 + 14v_2$$ So, $$D + 7DA = (3v_1 + v_2) + (7v_1 + 14v_2) = 10v_1 + 15v_2$$ **Note:** The user states $D + 7DA = -5v_1 + 22v_2$ which contradicts the calculation; the correct sum is $10v_1 + 15v_2$. --- **Final answers:** - $A = 4v_1 + 3v_2$ - $B = -3v_1$ - $C = -2v_1 - v_2$ - $D = 3v_1 + v_2$ - $AB = -7v_1 - 3v_2$ - $BA - CA = v_1 - v_2$ - $C + 7v_2 = -2v_1 + 6v_2$ - $D + 7DA = 10v_1 + 15v_2$