IF `barx times barb=barc times barb` and `barx . bara` =0 then `barx=`

Options

A. `(barb times (bara times barc))/(barb.barc)`
B. `((barb times barc) times bara)/(barb.bara)`
C. `((bara.(barc times barb))/(bara.barb)`
D. 0

Answer

To solve the problem step by step, we start with the given conditions:

1. **Given Conditions**:
- \( \bar{x} \times \bar{b} = \bar{c} \times \bar{b} \)
- \( \bar{x} \cdot \bar{a} = 0 \)

2. **Step 1: Cross Product Equality**:
We can rewrite the first condition:
\[ \bar{x} \times \bar{b} - \bar{c} \times \bar{b} = \bar{0} \]
This implies that the vectors \( \bar{x} \) and \( \bar{c} \) have the same cross product with \( \bar{b} \).

3. **Step 2: Apply the Vector Triple Product Identity**:
We can apply the vector triple product identity:
\[ \bar{a} \times (\bar{x} \times \bar{b}) = (\bar{a} \cdot \bar{b}) \bar{x} - (\bar{a} \cdot \bar{x}) \bar{b} \]
Therefore, we have:
\[ \bar{a} \times (\bar{x} \times \bar{b}) = \bar{a} \times (\bar{c} \times \bar{b}) \]

4. **Step 3: Expand Both Sides**:
Expanding both sides using the identity:
\[ (\bar{a} \cdot \bar{b}) \bar{x} - (\bar{a} \cdot \bar{x}) \bar{b} = (\bar{a} \cdot \bar{b}) \bar{c} - (\bar{a} \cdot \bar{c}) \bar{b} \]

5. **Step 4: Substitute \( \bar{x} \cdot \bar{a} = 0 \)**:
Since \( \bar{x} \cdot \bar{a} = 0 \), we can substitute this into our equation:
\[ (\bar{a} \cdot \bar{b}) \bar{x} = (\bar{a} \cdot \bar{b}) \bar{c} - (\bar{a} \cdot \bar{c}) \bar{b} \]

6. **Step 5: Solve for \( \bar{x} \)**:
Rearranging gives us:
\[ \bar{x} = \frac{(\bar{a} \cdot \bar{c}) \bar{b} - (\bar{a} \cdot \bar{b}) \bar{c}}{\bar{a} \cdot \bar{b}} \]

7. **Step 6: Simplify**:
We can express \( \bar{x} \) in terms of a cross product:
\[ \bar{x} = \frac{\bar{a} \times \bar{c} \times \bar{b}}{\bar{a} \cdot \bar{b}} \]

8. **Final Expression**:
Thus, we arrive at:
\[ \bar{x} = \frac{\bar{c} \times \bar{b} \times \bar{a}}{\bar{b} \cdot \bar{a}} \]