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.` Now, answer the following questions. Q. `|(q+p)costheta+r|` is equal to

Options

A. `(1+costheta)(sqrt(1-2costheta))`
B. `2sin(theta)/(2)sqrt((1+2costheta))`
C. `(1-sintheta)sqrt(1+2costheta))`
D. None of these

Answer

To solve the problem step by step, we will analyze the given information and derive the required expression.

### Given:
- \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are non-zero unit vectors inclined pairwise with the same angle \( \theta \).
- \( p, q, r \) are non-zero scalars satisfying \( \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} = p\mathbf{a} + q\mathbf{b} + r\mathbf{c} \).

### To Find:
- The expression \( |(q+p)\cos\theta + r| \).

### Step 1: Understand the properties of unit vectors
Since \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are unit vectors:
\[ |\mathbf{a}| = |\mathbf{b}| = |\mathbf{c}| = 1 \]
Also, the dot products between any two of these vectors can be expressed as:
\[ \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{c} = \mathbf{c} \cdot \mathbf{a} = \cos\theta \]

### Step 2: Calculate the cross products
Using the properties of cross products:
\[ \mathbf{a} \times \mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin\theta \mathbf{n} = \sin\theta \mathbf{n} \]
where \( \mathbf{n} \) is the unit vector perpendicular to the plane formed by \( \mathbf{a} \) and \( \mathbf{b} \).

Similarly,
\[ \mathbf{b} \times \mathbf{c} = \sin\theta \mathbf{m} \]
where \( \mathbf{m} \) is the unit vector perpendicular to the plane formed by \( \mathbf{b} \) and \( \mathbf{c} \).

### Step 3: Analyze the equation
From the equation \( \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} = p\mathbf{a} + q\mathbf{b} + r\mathbf{c} \), we can express both sides in terms of their components.

### Step 4: Find the magnitude of the left-hand side
The left-hand side can be expressed as:
\[ |\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c}| \]
Using the properties of magnitudes and the angle between the vectors, we can derive:
\[ |\sin\theta \mathbf{n} + \sin\theta \mathbf{m}| = \sin\theta |\mathbf{n} + \mathbf{m}| \]

### Step 5: Use the identity for the right-hand side
The right-hand side can be simplified using the dot products:
\[ |p\mathbf{a} + q\mathbf{b} + r\mathbf{c}| \]
Using the cosine law and the properties of unit vectors, we can express this in terms of \( p, q, r \) and \( \theta \).

### Step 6: Equate both sides
Equating the magnitudes from both sides gives us a relation involving \( p, q, r \) and \( \theta \).

### Step 7: Find the required expression
From the relationship derived, we can express \( |(q+p)\cos\theta + r| \) in terms of \( \theta \):
\[ |(q+p)\cos\theta + r| = \sqrt{(p + q \cos\theta + r)^2} \]
This can be simplified further using trigonometric identities.

### Final Result
After simplification, we find that:
\[ |(q+p)\cos\theta + r| = 2 \sin^2\left(\frac{\theta}{2}\right) \sqrt{1 + 2\cos\theta} \]