p,q and r are three vectors defined by `p=axx(b+c),q=bxx(c+a) and r=cxx(a+b)`
Statement I: p,q and r are coplanar.
Statement II: Vectors p,q,r are linearly independent.

Options

A. Both statement I and statement II are correct and statement II is the correct explanation of statement I
B. both statement I and statement II are correct but statement II is not the correct explanation of statement I
C. Statement I is correct but statement II is incorrect
D. Statement II is correct but statement I is incorrect

Answer

To determine the validity of the statements regarding the vectors \( p, q, \) and \( r \), we will analyze each statement step by step.

### Given Vectors:
1. \( p = a \times (b + c) \)
2. \( q = b \times (c + a) \)
3. \( r = c \times (a + b) \)

### Step 1: Express the vectors in terms of cross products
We can rewrite the vectors as:
- \( p = a \times b + a \times c \)
- \( q = b \times c + b \times a \)
- \( r = c \times a + c \times b \)

### Step 2: Check if the vectors are coplanar
Vectors \( p, q, r \) are coplanar if the scalar triple product \( p \cdot (q \times r) = 0 \).

### Step 3: Calculate \( q \times r \)
Using the distributive property of the cross product:
\[ q \times r = (b \times c + b \times a) \times (c \times a + c \times b) \]
Expanding this using the distributive property:
\[ = (b \times c) \times (c \times a) + (b \times c) \times (c \times b) + (b \times a) \times (c \times a) + (b \times a) \times (c \times b) \]

### Step 4: Use the vector triple product identity
Using the identity \( x \times (y \times z) = (x \cdot z)y - (x \cdot y)z \):
- The term \( (b \times c) \times (c \times a) \) simplifies to \( (b \cdot a)c - (b \cdot c)a \).
- The term \( (b \times c) \times (c \times b) \) is zero since \( c \times b \) is parallel to \( b \times c \).
- The term \( (b \times a) \times (c \times a) \) simplifies to \( (b \cdot a)c - (b \cdot c)a \).
- The term \( (b \times a) \times (c \times b) \) simplifies to \( (b \cdot b)c - (b \cdot c)b \) which is also zero.

### Step 5: Combine results
From the above calculations, we can see that:
\[ q \times r = (b \cdot a)c - (b \cdot c)a + 0 + 0 \]
Thus, we have:
\[ q \times r = (b \cdot a)c - (b \cdot c)a \]

### Step 6: Calculate \( p \cdot (q \times r) \)
Now we need to calculate \( p \cdot (q \times r) \):
\[ p \cdot (q \times r) = (a \times (b + c)) \cdot ((b \cdot a)c - (b \cdot c)a) \]
Using the property of dot and cross products, we can see that if \( p \) is orthogonal to \( q \times r \), then \( p \cdot (q \times r) = 0 \).

### Conclusion for Statement I
Since \( p \cdot (q \times r) = 0 \), the vectors \( p, q, r \) are coplanar. Therefore, Statement I is **true**.

### Conclusion for Statement II
Since we found that \( p, q, r \) are coplanar, they cannot be linearly independent. Thus, Statement II is **false**.

### Final Answer
- Statement I: True (Vectors \( p, q, r \) are coplanar)
- Statement II: False (Vectors \( p, q, r \) are not linearly independent)