IF the non-zero vectors `bara and barb` are perpendiculars to each other than the solution of the equation `barr times bara=barb` is given by

Options

A. `barr=xbara+1/(bara.bara) (bara times barb),x in R`
B. `barr=xbarb+1/(barb.barb) (bara times barb),x in R`
C. `barr=xbara times barb,x in R`
D. None of these

Answer

To solve the equation \( \mathbf{R} \times \mathbf{A} = \mathbf{B} \) given that the vectors \( \mathbf{A} \) and \( \mathbf{B} \) are non-zero and perpendicular to each other, we can follow these steps:

### Step 1: Cross both sides with vector \( \mathbf{A} \)
We start with the equation:
\[ \mathbf{R} \times \mathbf{A} = \mathbf{B} \]
Now, we cross both sides with vector \( \mathbf{A} \):
\[ (\mathbf{R} \times \mathbf{A}) \times \mathbf{A} = \mathbf{B} \times \mathbf{A} \]

### Step 2: Apply the vector triple product identity
Using the vector triple product identity, we have:
\[ \mathbf{X} \times \mathbf{Y} \times \mathbf{Z} = (\mathbf{X} \cdot \mathbf{Z}) \mathbf{Y} - (\mathbf{Y} \cdot \mathbf{Z}) \mathbf{X} \]
Applying this to our equation gives:
\[ (\mathbf{R} \cdot \mathbf{A}) \mathbf{A} - (\mathbf{A} \cdot \mathbf{A}) \mathbf{R} = \mathbf{B} \times \mathbf{A} \]

### Step 3: Simplify the equation
Let \( |\mathbf{A}|^2 = \mathbf{A} \cdot \mathbf{A} \). The equation simplifies to:
\[ (\mathbf{R} \cdot \mathbf{A}) \mathbf{A} - |\mathbf{A}|^2 \mathbf{R} = \mathbf{B} \times \mathbf{A} \]
Rearranging gives:
\[ |\mathbf{A}|^2 \mathbf{R} = (\mathbf{R} \cdot \mathbf{A}) \mathbf{A} - \mathbf{B} \times \mathbf{A} \]

### Step 4: Solve for \( \mathbf{R} \)
Now, we can express \( \mathbf{R} \):
\[ \mathbf{R} = \frac{1}{|\mathbf{A}|^2} \left( (\mathbf{R} \cdot \mathbf{A}) \mathbf{A} - \mathbf{B} \times \mathbf{A} \right) \]

### Step 5: Analyze the dot product
Next, we take the dot product of both sides of the original equation \( \mathbf{R} \times \mathbf{A} = \mathbf{B} \) with \( \mathbf{B} \):
\[ \mathbf{B} \cdot (\mathbf{R} \times \mathbf{A}) = 0 \]
Using the property of dot and cross products, we can express this as:
\[ (\mathbf{B} \cdot \mathbf{R}) |\mathbf{A}|^2 - (\mathbf{A} \cdot \mathbf{B}) \mathbf{R} = 0 \]
Since \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular, \( \mathbf{A} \cdot \mathbf{B} = 0 \), which leads to:
\[ \mathbf{B} \cdot \mathbf{R} = 0 \]
This indicates that \( \mathbf{R} \) is perpendicular to \( \mathbf{B} \).

### Step 6: Conclusion
Thus, the solution for \( \mathbf{R} \) can be expressed in terms of \( \mathbf{A} \) and \( \mathbf{B} \):
\[ \mathbf{R} = x \mathbf{A} + \frac{1}{|\mathbf{A}|^2} (\mathbf{A} \times \mathbf{B}) \]
where \( x \) is a scalar.