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 the scalar triple product \([\vec{i}, \vec{m}, \vec{n}] = 4\).

### Step 1: Write down the scalar triple product
The scalar triple product \([\vec{i}, \vec{m}, \vec{n}]\) can be expressed as:

\[ \vec{i} \cdot (\vec{m} \times \vec{n}) = 4 \]

This is our first equation.

### Step 2: Take the dot product of \(\vec{r}\) with \(\vec{i}\)
Now, we take the dot product of both sides of the equation for \(\vec{r}\) with \(\vec{i}\):

\[ \vec{r} \cdot \vec{i} = a(\vec{m} \times \vec{n}) \cdot \vec{i} + b(\vec{n} \times \vec{i}) \cdot \vec{i} + c(\vec{i} \times \vec{m}) \cdot \vec{i} \]

### Step 3: Simplify the dot products
Since \((\vec{n} \times \vec{i})\) and \((\vec{i} \times \vec{m})\) are perpendicular to \(\vec{i}\), their dot products with \(\vec{i}\) will be zero:

\[ \vec{r} \cdot \vec{i} = a(\vec{m} \times \vec{n}) \cdot \vec{i} \]

Using the first equation, we can substitute:

\[ \vec{r} \cdot \vec{i} = a \cdot 4 \]

### Step 4: Take the dot product of \(\vec{r}\) with \(\vec{m}\)
Next, we take the dot product of \(\vec{r}\) with \(\vec{m}\):

\[ \vec{r} \cdot \vec{m} = a(\vec{m} \times \vec{n}) \cdot \vec{m} + b(\vec{n} \times \vec{i}) \cdot \vec{m} + c(\vec{i} \times \vec{m}) \cdot \vec{m} \]

Again, \((\vec{m} \times \vec{n}) \cdot \vec{m}\) and \((\vec{i} \times \vec{m}) \cdot \vec{m}\) will be zero:

\[ \vec{r} \cdot \vec{m} = b(\vec{n} \times \vec{i}) \cdot \vec{m} \]

### Step 5: Take the dot product of \(\vec{r}\) with \(\vec{n}\)
Now, we take the dot product of \(\vec{r}\) with \(\vec{n}\):

\[ \vec{r} \cdot \vec{n} = a(\vec{m} \times \vec{n}) \cdot \vec{n} + b(\vec{n} \times \vec{i}) \cdot \vec{n} + c(\vec{i} \times \vec{m}) \cdot \vec{n} \]

Again, \((\vec{m} \times \vec{n}) \cdot \vec{n}\) and \((\vec{n} \times \vec{i}) \cdot \vec{n}\) will be zero:

\[ \vec{r} \cdot \vec{n} = c(\vec{i} \times \vec{m}) \cdot \vec{n} \]

### Step 6: Combine the results
Now we can combine the results from the dot products:

\[ \vec{r} \cdot \vec{i} + \vec{r} \cdot \vec{m} + \vec{r} \cdot \vec{n} = 4a + 4b + 4c = 4(a + b + c) \]

### Step 7: Substitute back into the desired expression
Now, we substitute this into the expression we want to evaluate:

\[ \frac{a + b + c}{\vec{r} \cdot (\vec{i} + \vec{m} + \vec{n})} = \frac{a + b + c}{4(a + b + c)} \]

### Step 8: Simplify the expression
This simplifies to:

\[ \frac{1}{4} \]

### Final Answer
Thus, the value of \(\frac{a + b + c}{\vec{r} \cdot (\vec{i} + \vec{m} + \vec{n})}\) is:

\[ \frac{1}{4} \]