Let a, b, c are non-zero unit vectors inclined pairwise with the same angle `theta`, p, q, r are non zero scalars satisfying `atimesb+btimesc=pa+qb+rc` Q. Volume of parallelopiped with edges a, b, c is

Options

A. `p+(q+r)costheta`
B. `(p+q+r)costheta`
C. `2p-(q+r)costheta`
D. None of these

Answer

To find the volume of the parallelepiped formed by the vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \), we can follow these steps:

### Step 1: Understand the properties of the vectors
Given that \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are non-zero unit vectors, we have:
\[ |\mathbf{a}| = |\mathbf{b}| = |\mathbf{c}| = 1 \]
Also, since they are inclined pairwise with the same angle \( \theta \), we can express their dot products as:
\[ \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{c} = \mathbf{c} \cdot \mathbf{a} = \cos \theta \]

### Step 2: Write the given equation
We are given the equation:
\[ \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} = p\mathbf{a} + q\mathbf{b} + r\mathbf{c} \]
Let’s denote this as Equation (1).

### Step 3: Take the dot product of Equation (1) with \( \mathbf{a} \)
Taking the dot product of both sides of Equation (1) with \( \mathbf{a} \):
\[ \mathbf{a} \cdot (\mathbf{a} \times \mathbf{b}) + \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = p(\mathbf{a} \cdot \mathbf{a}) + q(\mathbf{a} \cdot \mathbf{b}) + r(\mathbf{a} \cdot \mathbf{c}) \]
Using the property that \( \mathbf{a} \cdot (\mathbf{a} \times \mathbf{b}) = 0 \) (since the dot product of a vector with a vector perpendicular to it is zero), we simplify:
\[ 0 + \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = p(1) + q(\cos \theta) + r(\cos \theta) \]
Thus, we have:
\[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = p + (q + r) \cos \theta \]

### Step 4: Recognize the scalar triple product
The left-hand side \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) represents the volume \( V \) of the parallelepiped formed by the vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \):
\[ V = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \]

### Step 5: Final expression for volume
From our previous result, we can express the volume as:
\[ V = p + (q + r) \cos \theta \]

### Conclusion
The volume of the parallelepiped with edges \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) is given by:
\[ V = p + (q + r) \cos \theta \]