If `vecr=a(vecm xx vecn)+b(vecn xx vecI)+c(vecI xx vecm) and [vecI vecm vecn]=4," find "(a+b+c)/(vecr*(vecI+vecm+vecn))` :

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. 1
D. 2

Answer

To solve the problem, we need to find the value of \((a + b + c) / ( \vec{r} \cdot (\vec{I} + \vec{m} + \vec{n}))\) given that \(\vec{r} = a(\vec{m} \times \vec{n}) + b(\vec{n} \times \vec{I}) + c(\vec{I} \times \vec{m})\) and \([\vec{I}, \vec{m}, \vec{n}] = 4\).

### Step-by-step Solution:

1. **Understanding the Given Expression**:
We have \(\vec{r} = a(\vec{m} \times \vec{n}) + b(\vec{n} \times \vec{I}) + c(\vec{I} \times \vec{m})\).
We need to compute \(\vec{r} \cdot (\vec{I} + \vec{m} + \vec{n})\).

2. **Dot Product with \(\vec{I}\)**:
Calculate \(\vec{r} \cdot \vec{I}\):
\[ \vec{r} \cdot \vec{I} = a(\vec{m} \times \vec{n}) \cdot \vec{I} + b(\vec{n} \times \vec{I}) \cdot \vec{I} + c(\vec{I} \times \vec{m}) \cdot \vec{I} \]
The second and third terms vanish because the dot product of any vector with itself is zero:
\[ \vec{r} \cdot \vec{I} = a(\vec{m} \times \vec{n}) \cdot \vec{I} \]

3. **Using the Box Notation**:
The term \(a(\vec{m} \times \vec{n}) \cdot \vec{I}\) can be expressed using the box notation:
\[ \vec{r} \cdot \vec{I} = a[\vec{I}, \vec{m}, \vec{n}] \]
Given \([\vec{I}, \vec{m}, \vec{n}] = 4\):
\[ \vec{r} \cdot \vec{I} = 4a \]

4. **Dot Product with \(\vec{m}\)**:
Now calculate \(\vec{r} \cdot \vec{m}\):
\[ \vec{r} \cdot \vec{m} = a(\vec{m} \times \vec{n}) \cdot \vec{m} + b(\vec{n} \times \vec{I}) \cdot \vec{m} + c(\vec{I} \times \vec{m}) \cdot \vec{m} \]
The first and third terms vanish:
\[ \vec{r} \cdot \vec{m} = b(\vec{n} \times \vec{I}) \cdot \vec{m} = 4b \]

5. **Dot Product with \(\vec{n}\)**:
Now calculate \(\vec{r} \cdot \vec{n}\):
\[ \vec{r} \cdot \vec{n} = a(\vec{m} \times \vec{n}) \cdot \vec{n} + b(\vec{n} \times \vec{I}) \cdot \vec{n} + c(\vec{I} \times \vec{m}) \cdot \vec{n} \]
The first and second terms vanish:
\[ \vec{r} \cdot \vec{n} = c(\vec{I} \times \vec{m}) \cdot \vec{n} = 4c \]

6. **Summing the Dot Products**:
Now sum the results:
\[ \vec{r} \cdot (\vec{I} + \vec{m} + \vec{n}) = \vec{r} \cdot \vec{I} + \vec{r} \cdot \vec{m} + \vec{r} \cdot \vec{n} = 4a + 4b + 4c = 4(a + b + c) \]

7. **Final Expression**:
We need to find:
\[ \frac{a + b + c}{\vec{r} \cdot (\vec{I} + \vec{m} + \vec{n})} = \frac{a + b + c}{4(a + b + c)} \]
This simplifies to:
\[ \frac{1}{4} \]

### Final Answer:
\[ \frac{1}{4} \]