If the vectors `a hati + 3 hatj - 2 hatk and 3 hati - 4 hatj + b hatk` are collinear, then (a,b) =

Options

A. `((9)/(4) , (8)/(3))`
B. `(- (9)/(4), (8)/(3))`
C. `((9)/(4), - (8)/(3))`
D. `(- (9)/(4), - (8)/(3))`

Answer

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

### Step 1: Set up the ratio of the components

Since the vectors are collinear, we can express this relationship as:

\[ \frac{a}{3} = \frac{3}{-4} = \frac{-2}{b} \]

### Step 2: Solve for \( a \)

From the first part of the ratio:

\[ \frac{a}{3} = \frac{3}{-4} \]

Cross-multiplying gives:

\[ a \cdot (-4) = 3 \cdot 3 \]

\[ -4a = 9 \]

Dividing both sides by -4:

\[ a = -\frac{9}{4} \]

### Step 3: Solve for \( b \)

Now, using the second part of the ratio:

\[ \frac{-2}{b} = \frac{3}{-4} \]

Cross-multiplying gives:

\[ -2 \cdot (-4) = 3 \cdot b \]

\[ 8 = 3b \]

Dividing both sides by 3:

\[ b = \frac{8}{3} \]

### Conclusion

Thus, the values of \( (a, b) \) are:

\[ (a, b) = \left(-\frac{9}{4}, \frac{8}{3}\right) \]

### Final Answer

\[ (a, b) = \left(-\frac{9}{4}, \frac{8}{3}\right) \]

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