Let `alpha = (lambda-2) a + b` and `beta =(4lambda -2)a + 3b` be two given vectors where vectors a and b are non-collinear. The value of `lambda` for which vectors `alpha` and `beta` are collinear, is.

Options

A. 4
B. `-3`
C. 3
D. `-4`

Answer

To find the value of \( \lambda \) for which the vectors \( \alpha \) and \( \beta \) are collinear, we start with the definitions of the vectors:

Given:
\[ \alpha = (\lambda - 2) \mathbf{a} + \mathbf{b} \]
\[ \beta = (4\lambda - 2) \mathbf{a} + 3\mathbf{b} \]

### Step 1: Set up the condition for collinearity
Vectors \( \alpha \) and \( \beta \) are collinear if there exists a scalar \( k \) such that:
\[ \alpha = k \beta \]

This implies that the coefficients of \( \mathbf{a} \) and \( \mathbf{b} \) in both vectors must be proportional. Therefore, we can set up the following ratio:
\[ \frac{\lambda - 2}{4\lambda - 2} = \frac{1}{3} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3(\lambda - 2) = 1(4\lambda - 2) \]

### Step 3: Expand both sides
Expanding both sides results in:
\[ 3\lambda - 6 = 4\lambda - 2 \]

### Step 4: Rearrange the equation
Rearranging the equation to isolate \( \lambda \):
\[ 3\lambda - 4\lambda = -2 + 6 \]
\[ -\lambda = 4 \]

### Step 5: Solve for \( \lambda \)
Multiplying both sides by -1 gives:
\[ \lambda = -4 \]

### Conclusion
Thus, the value of \( \lambda \) for which the vectors \( \alpha \) and \( \beta \) are collinear is:
\[ \lambda = -4 \]

---