The problem states that:

$(a - 2)\mathbf{m} = (3a + b)\mathbf{n}$

We need to solve for $a$ and $b$. Here’s the step-by-step solution:

1. Equation Analysis:

   • The equation is vector-based, so we assume the vectors $\mathbf{m}$ and $\mathbf{n}$ are independent.
   • This implies that their coefficients must be equal on both sides of the equation.

2. Separate into Components:

   • The coefficient of $\mathbf{m}$ is $a - 2$.
   • The coefficient of $\mathbf{n}$ is $3a + b$.
   • Since $\mathbf{m}$ and $\mathbf{n}$ are independent, both coefficients must equal 0.

   This gives two equations: $a - 2 = 0$ $3a + b = 0$

3. Solve the System of Equations:

   • From the first equation: $a = 2$
   • Substitute $a = 2$ into the second equation: $3(2) + b = 0$ $6 + b = 0$ $b = -6$

4. Final Answer: $a = 2, \, b = -6$

Let me know if you'd like additional details or explanations! Here are 5 related questions for further exploration:

1. What does it mean for vectors to be independent in this context?
2. How would the solution change if $\mathbf{m}$ and $\mathbf{n}$ were dependent?
3. Can you visualize this problem geometrically in terms of the vectors $\mathbf{OA}$ and $\mathbf{OB}$?
4. How do you solve similar equations involving more vectors?
5. What is the importance of coefficients in vector equations?

Tip: Always check for vector independence when solving these types of equations to ensure