For what value of 'a' the vectors :
`2hat(i)-3hat(j)+4hat(k)` and `a hat(i)+6hat(j)-8hat(k)` are collinear ?

Answer

To determine the value of 'a' for which the vectors \( \mathbf{A} = 2\hat{i} - 3\hat{j} + 4\hat{k} \) and \( \mathbf{B} = a\hat{i} + 6\hat{j} - 8\hat{k} \) are collinear, we can follow these steps:

### Step 1: Understand the condition for collinearity
Two vectors \( \mathbf{A} \) and \( \mathbf{B} \) are collinear if there exists a scalar \( \lambda \) such that:
\[ \mathbf{B} = \lambda \mathbf{A} \]
This implies that the components of the vectors must satisfy the following ratios:
\[ \frac{B_x}{A_x} = \frac{B_y}{A_y} = \frac{B_z}{A_z} \]

### Step 2: Identify components of the vectors
From the given vectors:
- \( \mathbf{A} = 2\hat{i} - 3\hat{j} + 4\hat{k} \) has components \( A_x = 2, A_y = -3, A_z = 4 \)
- \( \mathbf{B} = a\hat{i} + 6\hat{j} - 8\hat{k} \) has components \( B_x = a, B_y = 6, B_z = -8 \)

### Step 3: Set up the equations based on the ratios
Using the collinearity condition, we can set up the following equations:
\[ \frac{a}{2} = \frac{6}{-3} = \frac{-8}{4} \]

### Step 4: Solve the first ratio
Calculating \( \frac{6}{-3} \):
\[ \frac{6}{-3} = -2 \]
Thus, we have:
\[ \frac{a}{2} = -2 \]

### Step 5: Solve for 'a'
To find 'a', we multiply both sides of the equation by 2:
\[ a = -2 \times 2 = -4 \]

### Step 6: Verify with the second ratio
Now, we check the second ratio \( \frac{-8}{4} \):
\[ \frac{-8}{4} = -2 \]
This confirms that both ratios are equal, validating our solution.

### Final Answer
The value of \( a \) for which the vectors are collinear is:
\[ \boxed{-4} \]