If `A =a_(1) hat"i"+ b_(1)hat"j"and B=a_(2)hat"i"+b_(2)hat"j"` the condition that they are perpendicula to each other is

Options

A. `(a_(1) )/(b_(1))=-(b_(2))/(a_(2))`
B. `a_(1)b_(1)= a_(2) b_(2)`
C. `(a_(1) )/(b_(1))=-(b_(1))/(b_(2))`
D. None of these

Explanation

To determine the condition under which the vectors \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Vectors**:
Given the vectors:
\[ \mathbf{A} = a_1 \hat{i} + b_1 \hat{j} \]
\[ \mathbf{B} = a_2 \hat{i} + b_2 \hat{j} \]

2. **Dot Product of the Vectors**:
The dot product \( \mathbf{A} \cdot \mathbf{B} \) is calculated as follows:
\[ \mathbf{A} \cdot \mathbf{B} = (a_1 \hat{i} + b_1 \hat{j}) \cdot (a_2 \hat{i} + b_2 \hat{j}) \]
Using the properties of the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = a_1 a_2 + b_1 b_2 \]

3. **Condition for Perpendicularity**:
For the vectors to be perpendicular, their dot product must equal zero:
\[ a_1 a_2 + b_1 b_2 = 0 \]

4. **Rearranging the Equation**:
We can rearrange this equation to express one term in terms of the other:
\[ a_1 a_2 = -b_1 b_2 \]

5. **Expressing Ratios**:
From the equation \( a_1 a_2 = -b_1 b_2 \), we can express the ratios:
\[ \frac{a_1}{b_1} = -\frac{b_2}{a_2} \]

### Final Condition:
Thus, the condition for the vectors \( \mathbf{A} \) and \( \mathbf{B} \) to be perpendicular is:
\[ \frac{a_1}{b_1} = -\frac{b_2}{a_2} \]