Three vectors `vecA,vecB` and `vecC` add up to zero.Find which is false.

Options

A. `(AxxB)xxC` is not zero unless B, C are parallel
B. `(AxxB).C` is not zero unless B, C are parallel
C. If A, B, C define a plane, `(AxxB)xxC` is in that plane
D. `(AxxB).C=|A||B||C| rarr C^(2)=A^(2)+B^(2)`

Explanation

To solve the problem of determining which statement is false regarding three vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) that add up to zero, we can follow these steps:

### Step 1: Understand the Vector Addition
Given that the vectors add up to zero, we can express this mathematically as:
\[ \vec{A} + \vec{B} + \vec{C} = 0 \]
This implies that:
\[ \vec{C} = -(\vec{A} + \vec{B}) \]

### Step 2: Analyze the Cross Product
From the equation \(\vec{A} + \vec{B} + \vec{C} = 0\), we can take the cross product of both sides with \(\vec{B}\):
\[ (\vec{A} + \vec{B} + \vec{C}) \times \vec{B} = 0 \times \vec{B} \]
This simplifies to:
\[ \vec{A} \times \vec{B} + \vec{B} \times \vec{B} + \vec{C} \times \vec{B} = 0 \]
Since \(\vec{B} \times \vec{B} = 0\), we have:
\[ \vec{A} \times \vec{B} + \vec{C} \times \vec{B} = 0 \]
Thus, we can express this as:
\[ \vec{A} \times \vec{B} = -\vec{C} \times \vec{B} \]

### Step 3: Dot Product Analysis
Next, we can analyze the dot product \(\vec{A} \times \vec{B} \cdot \vec{C}\):
Using the relation we derived:
\[ \vec{A} \times \vec{B} \cdot \vec{C} = (-\vec{C} \times \vec{B}) \cdot \vec{C} \]
The dot product of a vector with a vector that is perpendicular to it (which \(\vec{C} \times \vec{B}\) is) is zero:
\[ \vec{C} \times \vec{B} \cdot \vec{C} = 0 \]
Thus, we conclude:
\[ \vec{A} \times \vec{B} \cdot \vec{C} = 0 \]

### Step 4: Evaluate the Statements
Now we can evaluate the given statements to find the false one:

1. **Statement 1**: True, as derived above.
2. **Statement 2**: False, because we found that \(\vec{A} \times \vec{B} \cdot \vec{C} = 0\) regardless of whether \(\vec{B}\) and \(\vec{C}\) are parallel.
3. **Statement 3**: True, as the cross product of two vectors is perpendicular to the plane formed by them.
4. **Statement 4**: False, as the relationship given does not hold true in general for arbitrary vectors.

### Conclusion
The false statements are:
- **Statement 2**: \(\vec{A} \times \vec{B} \cdot \vec{C} \neq 0\) unless \(\vec{B}\) and \(\vec{C}\) are parallel.
- **Statement 4**: \(\vec{A} \times \vec{B} \cdot \vec{C} = |\vec{A}||\vec{B}||\vec{C}|\) is not generally true.