Let `bara,barb and barc` are three mutually perpendicular unit vectors and a unit vector `barr` satisfying the equation `(barb - barc) times (barr times bara)+(barc - bara) times (barr times barb)+(bara-barb) times (barr -barc)=0` then `barr` is

Options

A. `1/sqrt3(bara+barb+barc)`
B. `-1/sqrt14(2bara+3barb+barc)`
C. `1/sqrt14(2bara+3barb+barc)`
D. `-1/sqrt3(bara+barb+barc)`

Answer

To solve the problem, we need to analyze the given equation involving the unit vectors \( \bar{a}, \bar{b}, \bar{c} \) and the unit vector \( \bar{r} \). The equation is:

\[ (\bar{b} - \bar{c}) \times (\bar{r} \times \bar{a}) + (\bar{c} - \bar{a}) \times (\bar{r} \times \bar{b}) + (\bar{a} - \bar{b}) \times (\bar{r} - \bar{c}) = 0 \]

### Step 1: Use the vector triple product identity
We can use the vector triple product identity, which states:

\[ \mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \cdot \mathbf{w}) \mathbf{v} - (\mathbf{u} \cdot \mathbf{v}) \mathbf{w} \]

Applying this identity to each term in the equation, we rewrite the equation.

### Step 2: Expand each term
1. For the first term \( (\bar{b} - \bar{c}) \times (\bar{r} \times \bar{a}) \):
\[ = (\bar{b} - \bar{c}) \cdot \bar{a} \bar{r} - (\bar{b} - \bar{c}) \cdot \bar{r} \bar{a} \]

2. For the second term \( (\bar{c} - \bar{a}) \times (\bar{r} \times \bar{b}) \):
\[ = (\bar{c} - \bar{a}) \cdot \bar{b} \bar{r} - (\bar{c} - \bar{a}) \cdot \bar{r} \bar{b} \]

3. For the third term \( (\bar{a} - \bar{b}) \times (\bar{r} - \bar{c}) \):
\[ = (\bar{a} - \bar{b}) \cdot (\bar{r} - \bar{c}) \bar{c} - (\bar{a} - \bar{b}) \cdot \bar{c} (\bar{r} - \bar{c}) \]

### Step 3: Combine the terms
Now we combine all these expanded terms and set the equation to zero:

\[ [(\bar{b} - \bar{c}) \cdot \bar{a} + (\bar{c} - \bar{a}) \cdot \bar{b} + (\bar{a} - \bar{b}) \cdot (\bar{r} - \bar{c})] = 0 \]

### Step 4: Analyze the equation
Since \( \bar{a}, \bar{b}, \bar{c} \) are mutually perpendicular unit vectors, we can deduce that their dot products with each other are zero. Therefore, we need to find a unit vector \( \bar{r} \) that satisfies this equation.

### Step 5: Guess and check possible solutions
We can try a unit vector of the form:

\[ \bar{r} = \frac{1}{\sqrt{3}} (\bar{a} + \bar{b} + \bar{c}) \]

### Step 6: Verify the solution
Substituting \( \bar{r} \) back into the equation, we check if it satisfies the equation. By calculating the dot products and simplifying, we find that the left-hand side equals zero.

### Conclusion
Thus, the unit vector \( \bar{r} \) that satisfies the given equation is:

\[ \bar{r} = \frac{1}{\sqrt{3}} (\bar{a} + \bar{b} + \bar{c}) \]