Let `veca,vecb,vecc` be three noncolanar vectors and `vecp,vecq,vecr` are vectors defined by the relations` vecp= (vecbxxvecc)/([veca vecb vecc]), vecq= (veccxxveca)/([veca vecb vecc]), vecr= (vecaxxvecb)/([veca vecb vecc])` then the value of the expression `(veca+vecb).vecp+(vecb+vecc).vecq+(vecc+veca).vecr`. is equal to (A) 0 (B) 1 (C) 2 (D) 3

Options

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

Explanation

To solve the problem, we need to evaluate the expression \((\vec{a} + \vec{b}) \cdot \vec{p} + (\vec{b} + \vec{c}) \cdot \vec{q} + (\vec{c} + \vec{a}) \cdot \vec{r}\) given the definitions of \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\).

1. **Given Definitions**:
- \(\vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{a}, \vec{b}, \vec{c}]}\)
- \(\vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{a}, \vec{b}, \vec{c}]}\)
- \(\vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{a}, \vec{b}, \vec{c}]}\)

Here, \([\vec{a}, \vec{b}, \vec{c}]\) denotes the scalar triple product of the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).

2. **Evaluate Each Term**:
- **First Term**: \((\vec{a} + \vec{b}) \cdot \vec{p}\)
\[ = (\vec{a} + \vec{b}) \cdot \left(\frac{\vec{b} \times \vec{c}}{[\vec{a}, \vec{b}, \vec{c}]}\right) \]
\[ = \frac{(\vec{a} + \vec{b}) \cdot (\vec{b} \times \vec{c})}{[\vec{a}, \vec{b}, \vec{c}]} \]
Using the property of the scalar triple product, we know that \((\vec{a} + \vec{b}) \cdot (\vec{b} \times \vec{c}) = \vec{a} \cdot (\vec{b} \times \vec{c}) + \vec{b} \cdot (\vec{b} \times \vec{c})\). The second term is zero because the dot product of a vector with itself crossed with another vector is zero.
\[ = \frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{[\vec{a}, \vec{b}, \vec{c}]} \]
This is equal to 1.

- **Second Term**: \((\vec{b} + \vec{c}) \cdot \vec{q}\)
\[ = (\vec{b} + \vec{c}) \cdot \left(\frac{\vec{c} \times \vec{a}}{[\vec{a}, \vec{b}, \vec{c}]}\right) \]
\[ = \frac{(\vec{b} + \vec{c}) \cdot (\vec{c} \times \vec{a})}{[\vec{a}, \vec{b}, \vec{c}]} \]
Similarly, this can be simplified:
\[ = \frac{\vec{b} \cdot (\vec{c} \times \vec{a}) + \vec{c} \cdot (\vec{c} \times \vec{a})}{[\vec{a}, \vec{b}, \vec{c}]} \]
The second term is zero.
\[ = \frac{\vec{b} \cdot (\vec{c} \times \vec{a})}{[\vec{a}, \vec{b}, \vec{c}]} \]
This is also equal to 1.

- **Third Term**: \((\vec{c} + \vec{a}) \cdot \vec{r}\)
\[ = (\vec{c} + \vec{a}) \cdot \left(\frac{\vec{a} \times \vec{b}}{[\vec{a}, \vec{b}, \vec{c}]}\right) \]
\[ = \frac{(\vec{c} + \vec{a}) \cdot (\vec{a} \times \vec{b})}{[\vec{a}, \vec{b}, \vec{c}]} \]
Again, this can be simplified:
\[ = \frac{\vec{c} \cdot (\vec{a} \times \vec{b}) + \vec{a} \cdot (\vec{a} \times \vec{b})}{[\vec{a}, \vec{b}, \vec{c}]} \]
The second term is zero.
\[ = \frac{\vec{c} \cdot (\vec{a} \times \vec{b})}{[\vec{a}, \vec{b}, \vec{c}]} \]
This is equal to 1.

3. **Combine the Results**:
Now we can combine all the results:
\[ (\vec{a} + \vec{b}) \cdot \vec{p} + (\vec{b} + \vec{c}) \cdot \vec{q} + (\vec{c} + \vec{a}) \cdot \vec{r} = 1 + 1 + 1 = 3 \]

Thus, the value of the expression is \(3\).