The value of 'a' when the vectors :
`2hat(i)-3hat(j)+4hat(k)` and `ahat(i)+b hat(j)-8 hat(k)` are collinear is __________.

Answer

To find the value of 'a' when the vectors \( \mathbf{A} = 2\hat{i} - 3\hat{j} + 4\hat{k} \) and \( \mathbf{B} = a\hat{i} + b\hat{j} - 8\hat{k} \) are collinear, we can use the property that two vectors are collinear if the ratios of their corresponding components are equal.

### Step-by-Step Solution:

1. **Write the vectors**:
\[ \mathbf{A} = 2\hat{i} - 3\hat{j} + 4\hat{k} \]
\[ \mathbf{B} = a\hat{i} + b\hat{j} - 8\hat{k} \]

2. **Set up the ratios for collinearity**:
For the vectors to be collinear, the ratios of the corresponding components must be equal:
\[ \frac{2}{a} = \frac{-3}{b} = \frac{4}{-8} \]

3. **Calculate the ratio from the k-components**:
From the k-components, we have:
\[ \frac{4}{-8} = -\frac{1}{2} \]

4. **Set up equations using the ratios**:
Now we can set up equations using the ratios:
\[ \frac{2}{a} = -\frac{1}{2} \]
\[ \frac{-3}{b} = -\frac{1}{2} \]

5. **Solve for 'a'**:
From the first equation:
\[ 2 = -\frac{1}{2}a \implies a = -4 \]

6. **Conclusion**:
Thus, the value of \( a \) when the vectors are collinear is:
\[ \boxed{-4} \]