1. **Stating the problem:** We have four forces: $7\mathbf{i} - 2\mathbf{j}$, $-6\mathbf{i} + 5\mathbf{j}$, $3\mathbf{i} + 6\mathbf{j}$, and $a\mathbf{i} + b\mathbf{j}$. Their resultant is $11\mathbf{i} - 2\mathbf{j}$. We need to find $a$ and $b$. 2. **Formula and rules:** The resultant force is the vector sum of all forces. So, $$ (7 - 6 + 3 + a)\mathbf{i} + (-2 + 5 + 6 + b)\mathbf{j} = 11\mathbf{i} - 2\mathbf{j} $$ 3. **Equating components:** For the $\mathbf{i}$ components: $$ 7 - 6 + 3 + a = 11 $$ Simplify: $$ 4 + a = 11 $$ Subtract 4 from both sides: $$ \cancel{4} + a = 11 - \cancel{4} $$ $$ a = 7 $$ For the $\mathbf{j}$ components: $$ -2 + 5 + 6 + b = -2 $$ Simplify: $$ 9 + b = -2 $$ Subtract 9 from both sides: $$ \cancel{9} + b = -2 - \cancel{9} $$ $$ b = -11 $$ 4. **Answer for (a):** $$ a = 7, \quad b = -11 $$ 5. **For (b):** When a fifth force is added, equilibrium means the total force is zero: $$ 7\mathbf{i} - 2\mathbf{j} + (-6\mathbf{i} + 5\mathbf{j}) + (3\mathbf{i} + 6\mathbf{j}) + (7\mathbf{i} - 11\mathbf{j}) + \mathbf{F}_5 = \mathbf{0} $$ Sum the first four forces: $$ (7 - 6 + 3 + 7)\mathbf{i} + (-2 + 5 + 6 - 11)\mathbf{j} + \mathbf{F}_5 = \mathbf{0} $$ Simplify: $$ 11\mathbf{i} - 2\mathbf{j} + \mathbf{F}_5 = \mathbf{0} $$ Therefore, $$ \mathbf{F}_5 = -11\mathbf{i} + 2\mathbf{j} $$ **Final answers:** (a) $a=7$, $b=-11$ (b) The fifth force is $-11\mathbf{i} + 2\mathbf{j}$.