If `overline(a),overline(b), overline(c)` are three non-coplanar vectors and `overline(p), overline(q), overline(r)` are vectors defined by the relations `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)]])`, then `(overline(a)+overline(b))*overline(p)+(overline(b)+overline(c))*overline(q)+(overline(c)+overline(a))*overline(r)=`

Options

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

Answer

To solve the given problem step by step, we will use the definitions of the vectors \( \overline{p}, \overline{q}, \overline{r} \) and compute the expression \( (\overline{a} + \overline{b}) \cdot \overline{p} + (\overline{b} + \overline{c}) \cdot \overline{q} + (\overline{c} + \overline{a}) \cdot \overline{r} \).

### Step 1: Substitute the definitions of \( \overline{p}, \overline{q}, \overline{r} \)

We know:
\[ \overline{p} = \frac{\overline{b} \times \overline{c}}{[\overline{a}, \overline{b}, \overline{c}]}, \quad \overline{q} = \frac{\overline{c} \times \overline{a}}{[\overline{a}, \overline{b}, \overline{c}]}, \quad \overline{r} = \frac{\overline{a} \times \overline{b}}{[\overline{a}, \overline{b}, \overline{c}]} \]

Substituting these into the expression:
\[ (\overline{a} + \overline{b}) \cdot \overline{p} + (\overline{b} + \overline{c}) \cdot \overline{q} + (\overline{c} + \overline{a}) \cdot \overline{r} \]
becomes:
\[ (\overline{a} + \overline{b}) \cdot \left( \frac{\overline{b} \times \overline{c}}{[\overline{a}, \overline{b}, \overline{c}]} \right) + (\overline{b} + \overline{c}) \cdot \left( \frac{\overline{c} \times \overline{a}}{[\overline{a}, \overline{b}, \overline{c}]} \right) + (\overline{c} + \overline{a}) \cdot \left( \frac{\overline{a} \times \overline{b}}{[\overline{a}, \overline{b}, \overline{c}]} \right) \]

### Step 2: Factor out the common denominator

The common denominator is \( [\overline{a}, \overline{b}, \overline{c}] \). Thus, we can factor it out:
\[ \frac{1}{[\overline{a}, \overline{b}, \overline{c}]} \left[ (\overline{a} + \overline{b}) \cdot (\overline{b} \times \overline{c}) + (\overline{b} + \overline{c}) \cdot (\overline{c} \times \overline{a}) + (\overline{c} + \overline{a}) \cdot (\overline{a} \times \overline{b}) \right] \]

### Step 3: Expand each dot product

Now we will expand each term inside the brackets:
1. \( (\overline{a} + \overline{b}) \cdot (\overline{b} \times \overline{c}) = \overline{a} \cdot (\overline{b} \times \overline{c}) + \overline{b} \cdot (\overline{b} \times \overline{c}) \)
2. \( (\overline{b} + \overline{c}) \cdot (\overline{c} \times \overline{a}) = \overline{b} \cdot (\overline{c} \times \overline{a}) + \overline{c} \cdot (\overline{c} \times \overline{a}) \)
3. \( (\overline{c} + \overline{a}) \cdot (\overline{a} \times \overline{b}) = \overline{c} \cdot (\overline{a} \times \overline{b}) + \overline{a} \cdot (\overline{a} \times \overline{b}) \)

### Step 4: Simplify using properties of the dot and cross products

Using the property that the dot product of a vector with a cross product involving itself is zero:
- \( \overline{b} \cdot (\overline{b} \times \overline{c}) = 0 \)
- \( \overline{c} \cdot (\overline{c} \times \overline{a}) = 0 \)
- \( \overline{a} \cdot (\overline{a} \times \overline{b}) = 0 \)

Thus, we are left with:
\[ \overline{a} \cdot (\overline{b} \times \overline{c}) + \overline{b} \cdot (\overline{c} \times \overline{a}) + \overline{c} \cdot (\overline{a} \times \overline{b}) \]

### Step 5: Recognize the expression as a triple product

The expression \( \overline{a} \cdot (\overline{b} \times \overline{c}) + \overline{b} \cdot (\overline{c} \times \overline{a}) + \overline{c} \cdot (\overline{a} \times \overline{b}) \) is known to equal \( [\overline{a}, \overline{b}, \overline{c}] \).

### Step 6: Final result

Thus, we have:
\[ \frac{[\overline{a}, \overline{b}, \overline{c}]}{[\overline{a}, \overline{b}, \overline{c}]} = 1 \]

So the final answer is:
\[ 1 \]