If `p = a xx (b + c) + b xx (c + a) + c xx (a + b)`
`q = a xx (b xx c) + b xx (c xxa) + c xx (a xx b)`
`r = (a.b)^(2) + (a xx b)^(2)` then which one is incorrect

Options

A. `p = 0`
B. `q = 0`
C. `r = a^(2)b^(2)`
D. `r=0`

Answer

To solve the problem, we need to analyze the vectors \( p \), \( q \), and \( r \) given in the question and determine which one is incorrect.

### Step 1: Analyze vector \( p \)

The expression for \( p \) is given by:

\[ p = a \times (b + c) + b \times (c + a) + c \times (a + b) \]

We can expand this using the distributive property of the cross product:

\[ p = a \times b + a \times c + b \times c + b \times a + c \times a + c \times b \]

Now, we can rearrange the terms:

\[ p = (a \times b + b \times a) + (a \times c + c \times a) + (b \times c + c \times b) \]

Using the property of the cross product \( a \times b = - (b \times a) \), we can simplify:

\[ p = 0 + 0 + 0 = 0 \]

Thus, we conclude that:

\[ p = \mathbf{0} \]

### Step 2: Analyze vector \( q \)

The expression for \( q \) is given by:

\[ q = a \times (b \times c) + b \times (c \times a) + c \times (a \times b) \]

Using the vector triple product identity \( a \times (b \times c) = (a \cdot c) b - (a \cdot b) c \), we can expand each term:

1. \( a \times (b \times c) = (a \cdot c) b - (a \cdot b) c \)
2. \( b \times (c \times a) = (b \cdot a) c - (b \cdot c) a \)
3. \( c \times (a \times b) = (c \cdot b) a - (c \cdot a) b \)

Combining these gives:

\[ q = [(a \cdot c) b - (a \cdot b) c] + [(b \cdot a) c - (b \cdot c) a] + [(c \cdot b) a - (c \cdot a) b] \]

After simplification, we find that:

\[ q = 0 \]

### Step 3: Analyze vector \( r \)

The expression for \( r \) is given by:

\[ r = (a \cdot b)^2 + (a \times b)^2 \]

Using the identity \( (a \cdot b)^2 + (a \times b)^2 = |a|^2 |b|^2 \):

\[ r = |a|^2 |b|^2 \]

This is a non-negative quantity, and it is not equal to zero unless either \( a \) or \( b \) is the zero vector.

### Conclusion

From our analysis, we have:

- \( p = 0 \) (correct)
- \( q = 0 \) (correct)
- \( r = |a|^2 |b|^2 \) (not necessarily zero)

Thus, the incorrect statement is related to \( r \) since it is not equal to zero unless \( a \) or \( b \) is zero.

### Final Answer

The incorrect option is related to \( r \).