If `overline(p)=(overline(b)timesoverline(c))/([[overline(a), overline(b), overline(c)]]), overline(q)=(overline(c)timesoverline(a))/([[overline(a), overline(b), overline(c)]]), overline(r)=(overline(a)timesoverline(b))/([[overline(a), overline(b), overline(c)]])`, where `overline(a), overline(b), overline(c)` are three non-coplanar vectors, then `(overline(a)+overline(b)+overline(c))*(overline(p)+overline(q)+overline(r))=`

Options

A. `3`
B. `2`
C. `1`
D. `0`

Answer

To solve the problem, we need to compute the expression \((\overline{a} + \overline{b} + \overline{c}) \cdot (\overline{p} + \overline{q} + \overline{r})\) using the definitions of the vectors \(\overline{p}\), \(\overline{q}\), and \(\overline{r}\).

### Step 1: Write down the expression
We start with the expression we need to compute:
\[ (\overline{a} + \overline{b} + \overline{c}) \cdot (\overline{p} + \overline{q} + \overline{r}) \]

### Step 2: Substitute the definitions of \(\overline{p}\), \(\overline{q}\), and \(\overline{r}\)
Using the definitions given in the problem:
\[ \overline{p} = \frac{\overline{b} \times \overline{c}}{[\overline{a}, \overline{b}, \overline{c}]}, \quad \overline{q} = \frac{\overline{c} \times \overline{a}}{[\overline{a}, \overline{b}, \overline{c}]}, \quad \overline{r} = \frac{\overline{a} \times \overline{b}}{[\overline{a}, \overline{b}, \overline{c}]} \]
We can rewrite the expression as:
\[ (\overline{a} + \overline{b} + \overline{c}) \cdot \left( \frac{\overline{b} \times \overline{c}}{[\overline{a}, \overline{b}, \overline{c}]} + \frac{\overline{c} \times \overline{a}}{[\overline{a}, \overline{b}, \overline{c}]} + \frac{\overline{a} \times \overline{b}}{[\overline{a}, \overline{b}, \overline{c}]} \right) \]

### Step 3: Factor out the scalar denominator
The denominator is common in all three terms, so we can factor it out:
\[ = \frac{1}{[\overline{a}, \overline{b}, \overline{c}]} \left( (\overline{a} + \overline{b} + \overline{c}) \cdot (\overline{b} \times \overline{c}) + (\overline{a} + \overline{b} + \overline{c}) \cdot (\overline{c} \times \overline{a}) + (\overline{a} + \overline{b} + \overline{c}) \cdot (\overline{a} \times \overline{b}) \right) \]

### Step 4: Evaluate each dot product
1. **First term**:
\[ (\overline{a} + \overline{b} + \overline{c}) \cdot (\overline{b} \times \overline{c}) = \overline{a} \cdot (\overline{b} \times \overline{c}) + \overline{b} \cdot (\overline{b} \times \overline{c}) + \overline{c} \cdot (\overline{b} \times \overline{c}) \]
The second and third terms are zero because the dot product of a vector with a cross product involving itself is zero. Thus:
\[ = \overline{a} \cdot (\overline{b} \times \overline{c}) = [\overline{a}, \overline{b}, \overline{c}] \]

2. **Second term**:
\[ (\overline{a} + \overline{b} + \overline{c}) \cdot (\overline{c} \times \overline{a}) = \overline{a} \cdot (\overline{c} \times \overline{a}) + \overline{b} \cdot (\overline{c} \times \overline{a}) + \overline{c} \cdot (\overline{c} \times \overline{a}) \]
Again, the first and last terms are zero. Thus:
\[ = \overline{b} \cdot (\overline{c} \times \overline{a}) = [\overline{b}, \overline{c}, \overline{a}] \]

3. **Third term**:
\[ (\overline{a} + \overline{b} + \overline{c}) \cdot (\overline{a} \times \overline{b}) = \overline{a} \cdot (\overline{a} \times \overline{b}) + \overline{b} \cdot (\overline{a} \times \overline{b}) + \overline{c} \cdot (\overline{a} \times \overline{b}) \]
The first and second terms are zero. Thus:
\[ = \overline{c} \cdot (\overline{a} \times \overline{b}) = [\overline{c}, \overline{a}, \overline{b}] \]

### Step 5: Combine the results
Now we combine the results:
\[ = \frac{1}{[\overline{a}, \overline{b}, \overline{c}]} \left( [\overline{a}, \overline{b}, \overline{c}] + [\overline{b}, \overline{c}, \overline{a}] + [\overline{c}, \overline{a}, \overline{b}] \right) \]
Since all the scalar triple products are equal, we have:
\[ = \frac{3[\overline{a}, \overline{b}, \overline{c}]}{[\overline{a}, \overline{b}, \overline{c}]} = 3 \]

### Final Answer
Thus, the final result is:
\[ (\overline{a} + \overline{b} + \overline{c}) \cdot (\overline{p} + \overline{q} + \overline{r}) = 3 \]