If a and b are not perpendicular to each other and `r xx a = b xx a, r.c =0`, then r is equal to

Options

A. `a - c`
B. `b = xa` for all scalars x
C. `b - ((b.c)a)/((a.c))`
D. none of these

Answer

To solve the problem, we need to analyze the given conditions and derive the value of vector \( r \).

### Step-by-Step Solution:

1. **Given Conditions**:
- We have two vectors \( a \) and \( b \) that are not perpendicular to each other.
- The equations provided are:
\[ r \times a = b \times a \]
\[ r \cdot c = 0 \]

2. **Understanding the Cross Product**:
- The equation \( r \times a = b \times a \) implies that the vector \( r \) can be expressed in terms of \( b \) and some scalar multiple of \( a \).
- Rearranging gives:
\[ r \times a - b \times a = 0 \]
- This indicates that \( r - b \) is parallel to \( a \). Therefore, we can write:
\[ r = b + \lambda a \]
where \( \lambda \) is some scalar.

3. **Using the Dot Product Condition**:
- The second condition states \( r \cdot c = 0 \). Substituting \( r \) from the previous step gives:
\[ (b + \lambda a) \cdot c = 0 \]
- Expanding this, we have:
\[ b \cdot c + \lambda (a \cdot c) = 0 \]

4. **Solving for \( \lambda \)**:
- Rearranging the equation gives:
\[ \lambda (a \cdot c) = -b \cdot c \]
- Thus, we can express \( \lambda \) as:
\[ \lambda = -\frac{b \cdot c}{a \cdot c} \]
- This is valid as long as \( a \cdot c \neq 0 \).

5. **Final Expression for \( r \)**:
- Substituting \( \lambda \) back into the expression for \( r \):
\[ r = b - \frac{b \cdot c}{a \cdot c} a \]

### Conclusion:
Thus, the vector \( r \) can be expressed as:
\[ r = b - \frac{b \cdot c}{a \cdot c} a \]