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 equations:

1. **Given Equations:**
\[ \bar{x} \times \bar{b} = \bar{c} \times \bar{b} \]
\[ \bar{x} \cdot \bar{a} = 0 \]

2. **Cross Product Manipulation:**
From the first equation, we can rearrange it:
\[ \bar{x} \times \bar{b} - \bar{c} \times \bar{b} = 0 \]
This implies:
\[ (\bar{x} - \bar{c}) \times \bar{b} = 0 \]
This means that either \(\bar{x} - \bar{c} = 0\) or \(\bar{b} = 0\). Since \(\bar{b}\) cannot be zero (as it is a vector), we have:
\[ \bar{x} = \bar{c} \]

3. **Dot Product Condition:**
Now, we also have the condition:
\[ \bar{x} \cdot \bar{a} = 0 \]
Substituting \(\bar{x} = \bar{c}\) into this condition gives us:
\[ \bar{c} \cdot \bar{a} = 0 \]
This means that \(\bar{c}\) is orthogonal to \(\bar{a}\).

4. **Final Result:**
Therefore, we conclude that:
\[ \bar{x} = \bar{c} \]

### Summary of the Solution:
The solution shows that \(\bar{x}\) is equal to \(\bar{c}\) based on the conditions provided.