The sum, difference and cross produce of two vectors `A` and `B` are mutually perpendicular if

Options

A. `A` and `B` are perpendicular to each other and `|A|=|B|`
B. `A` and `B` are perpendicular to each other
C. `A` and `B` are perpendicular but their magnitude are arbitrary
D. `|A|=|B|` and their directions are arbitrary

Answer

To determine the conditions under which the sum, difference, and cross product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) are mutually perpendicular, we can follow these steps:

### Step 1: Define the vectors
Let:
- \( \mathbf{R_1} = \mathbf{A} + \mathbf{B} \)
- \( \mathbf{R_2} = \mathbf{A} - \mathbf{B} \)
- \( \mathbf{R_3} = \mathbf{A} \times \mathbf{B} \)

### Step 2: Set up the condition for perpendicularity
For the vectors \( \mathbf{R_1} \), \( \mathbf{R_2} \), and \( \mathbf{R_3} \) to be mutually perpendicular, the following conditions must hold:
- \( \mathbf{R_1} \cdot \mathbf{R_2} = 0 \)
- \( \mathbf{R_1} \cdot \mathbf{R_3} = 0 \)
- \( \mathbf{R_2} \cdot \mathbf{R_3} = 0 \)

### Step 3: Calculate the dot product \( \mathbf{R_1} \cdot \mathbf{R_2} \)
Calculating \( \mathbf{R_1} \cdot \mathbf{R_2} \):
\[ \mathbf{R_1} \cdot \mathbf{R_2} = (\mathbf{A} + \mathbf{B}) \cdot (\mathbf{A} - \mathbf{B}) = \mathbf{A} \cdot \mathbf{A} - \mathbf{B} \cdot \mathbf{B} = |\mathbf{A}|^2 - |\mathbf{B}|^2 \]
Setting this equal to zero gives:
\[ |\mathbf{A}|^2 - |\mathbf{B}|^2 = 0 \implies |\mathbf{A}| = |\mathbf{B}| \]

### Step 4: Analyze the direction of vectors
Next, we need to analyze the directions of \( \mathbf{A} \) and \( \mathbf{B} \). Since \( \mathbf{R_1} \) and \( \mathbf{R_2} \) are perpendicular, we can visualize this geometrically.

If we assume \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular, we can draw them such that:
- \( \mathbf{A} \) is along the x-axis.
- \( \mathbf{B} \) is along the y-axis.

This configuration will ensure that:
- \( \mathbf{R_1} \) and \( \mathbf{R_2} \) are at right angles to each other.

### Step 5: Conclusion
From the calculations and geometric analysis, we conclude that for the sum, difference, and cross product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) to be mutually perpendicular, the conditions are:
1. \( \mathbf{A} \) and \( \mathbf{B} \) must be perpendicular to each other.
2. The magnitudes of \( \mathbf{A} \) and \( \mathbf{B} \) must be equal.

Thus, the correct answer is that the sum, difference, and cross product of the vectors are mutually perpendicular if \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular to each other and \( |\mathbf{A}| = |\mathbf{B}| \).