If a and b are non-collinear vectors, then the value of a for which the vectors `u=(a - 2)a +b and V = (2+ 3a)a - 3b` are collinear, is

Options

A. `3/2`
B. `2/3`
C. `(-3)/2`
D. `(-2)/3`

Answer

To solve the problem, we need to find the value of \( a \) for which the vectors \( \mathbf{u} \) and \( \mathbf{v} \) are collinear. Two vectors are collinear if one is a scalar multiple of the other.

Given:
\[ \mathbf{u} = (a - 2)\mathbf{a} + \mathbf{b} \]
\[ \mathbf{v} = (2 + 3a)\mathbf{a} - 3\mathbf{b} \]

### Step 1: Set up the collinearity condition
For the vectors \( \mathbf{u} \) and \( \mathbf{v} \) to be collinear, we can express this as:
\[ \mathbf{u} = \lambda \mathbf{v} \]
for some scalar \( \lambda \).

### Step 2: Substitute the expressions for \( \mathbf{u} \) and \( \mathbf{v} \)
Substituting the expressions we have:
\[ (a - 2)\mathbf{a} + \mathbf{b} = \lambda \left((2 + 3a)\mathbf{a} - 3\mathbf{b}\right) \]

### Step 3: Expand the right-hand side
Expanding the right-hand side gives:
\[ (a - 2)\mathbf{a} + \mathbf{b} = \lambda(2 + 3a)\mathbf{a} - 3\lambda \mathbf{b} \]

### Step 4: Equate coefficients of \( \mathbf{a} \) and \( \mathbf{b} \)
From this equation, we can equate the coefficients of \( \mathbf{a} \) and \( \mathbf{b} \) on both sides:

1. Coefficient of \( \mathbf{a} \):
\[ a - 2 = \lambda(2 + 3a) \]

2. Coefficient of \( \mathbf{b} \):
\[ 1 = -3\lambda \]

### Step 5: Solve for \( \lambda \)
From the second equation:
\[ \lambda = -\frac{1}{3} \]

### Step 6: Substitute \( \lambda \) back into the first equation
Now substitute \( \lambda \) into the first equation:
\[ a - 2 = -\frac{1}{3}(2 + 3a) \]

### Step 7: Simplify the equation
Multiply both sides by \( -3 \) to eliminate the fraction:
\[ -3(a - 2) = 2 + 3a \]
\[ -3a + 6 = 2 + 3a \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ 6 - 2 = 3a + 3a \]
\[ 4 = 6a \]

### Step 9: Solve for \( a \)
Dividing both sides by 6:
\[ a = \frac{4}{6} = \frac{2}{3} \]

### Conclusion
Thus, the value of \( a \) for which the vectors \( \mathbf{u} \) and \( \mathbf{v} \) are collinear is:
\[ \boxed{\frac{2}{3}} \]