The value of 'a' so that the vectors `2 hat(i) - 3hat(j) + 4hat(k) and a hat(i) + 6hat(j) - 8hat(k)` are collinear

Options

A. 4
B. `-4`
C. 2
D. 0

Answer

To find the value of 'a' such that the vectors \( \mathbf{p} = 2\hat{i} - 3\hat{j} + 4\hat{k} \) and \( \mathbf{q} = a\hat{i} + 6\hat{j} - 8\hat{k} \) are collinear, we can follow these steps:

### Step 1: Understand the condition for collinearity
Two vectors \( \mathbf{p} \) and \( \mathbf{q} \) are collinear if there exists a scalar \( \lambda \) such that:
\[ \mathbf{p} = \lambda \mathbf{q} \]

### Step 2: Set up the equation
Using the vectors given:
\[ 2\hat{i} - 3\hat{j} + 4\hat{k} = \lambda (a\hat{i} + 6\hat{j} - 8\hat{k}) \]

### Step 3: Expand the right side
Expanding the right-hand side gives:
\[ 2\hat{i} - 3\hat{j} + 4\hat{k} = \lambda a \hat{i} + 6\lambda \hat{j} - 8\lambda \hat{k} \]

### Step 4: Equate components
Now we can equate the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) from both sides:
1. For \( \hat{i} \):
\[ 2 = \lambda a \quad \text{(1)} \]
2. For \( \hat{j} \):
\[ -3 = 6\lambda \quad \text{(2)} \]
3. For \( \hat{k} \):
\[ 4 = -8\lambda \quad \text{(3)} \]

### Step 5: Solve for \( \lambda \)
From equation (2):
\[ \lambda = \frac{-3}{6} = -\frac{1}{2} \]

### Step 6: Verify \( \lambda \) with equation (3)
Substituting \( \lambda = -\frac{1}{2} \) into equation (3):
\[ 4 = -8\left(-\frac{1}{2}\right) \implies 4 = 4 \quad \text{(True)} \]

### Step 7: Substitute \( \lambda \) back into equation (1)
Now substitute \( \lambda = -\frac{1}{2} \) into equation (1):
\[ 2 = -\frac{1}{2} a \]
Multiplying both sides by -2:
\[ -4 = a \]

### Conclusion
Thus, the value of \( a \) such that the vectors are collinear is:
\[ \boxed{-4} \]