If A,B,C,D are four points in space, then `|vec(AB)xvec(CD)+vec(BC)xxvec(AD)+vec(CA)xxvec(BD)|=k (are of /_\ABC) where k=` (A) 5 (B) 4 (C) 2 (D) none of these

Options

A. 2
B. 1
C. 3
D. 4

Explanation

To solve the problem, we need to analyze the expression given and relate it to the area of triangle ABC. Let's go through the steps systematically.

### Step 1: Define the vectors
Let the position vectors of points A, B, C, and D be represented as:
- \( \vec{A} \) for point A
- \( \vec{B} \) for point B
- \( \vec{C} \) for point C
- \( \vec{D} \) for point D

We can express the vectors as follows:
- \( \vec{AB} = \vec{B} - \vec{A} \)
- \( \vec{BC} = \vec{C} - \vec{B} \)
- \( \vec{CA} = \vec{A} - \vec{C} \)
- \( \vec{AD} = \vec{D} - \vec{A} \)
- \( \vec{CD} = \vec{D} - \vec{C} \)
- \( \vec{BD} = \vec{D} - \vec{B} \)

### Step 2: Write the expression
We need to evaluate the expression:
\[ |\vec{AB} \times \vec{CD} + \vec{BC} \times \vec{AD} + \vec{CA} \times \vec{BD}| \]

Substituting the vectors:
\[ |\vec{(B - A)} \times \vec{(D - C)} + \vec{(C - B)} \times \vec{(D - A)} + \vec{(A - C)} \times \vec{(D - B)}| \]

### Step 3: Expand the cross products
Using the properties of the cross product, we can expand each term:
1. \( \vec{AB} \times \vec{CD} = (\vec{B} - \vec{A}) \times (\vec{D} - \vec{C}) \)
2. \( \vec{BC} \times \vec{AD} = (\vec{C} - \vec{B}) \times (\vec{D} - \vec{A}) \)
3. \( \vec{CA} \times \vec{BD} = (\vec{A} - \vec{C}) \times (\vec{D} - \vec{B}) \)

### Step 4: Use the area of triangle ABC
The area of triangle ABC can be expressed as:
\[ \text{Area} = \frac{1}{2} |\vec{AB} \times \vec{AC}| \]
where \( \vec{AC} = \vec{C} - \vec{A} \).

### Step 5: Relate the expression to the area
We need to find \( k \) such that:
\[ |\vec{AB} \times \vec{CD} + \vec{BC} \times \vec{AD} + \vec{CA} \times \vec{BD}| = k \cdot \text{Area of } \triangle ABC \]

### Step 6: Simplify and find k
After performing the cross products and simplifying, we will find that:
\[ |\vec{AB} \times \vec{CD} + \vec{BC} \times \vec{AD} + \vec{CA} \times \vec{BD}| = 4 \cdot \text{Area of } \triangle ABC \]
Thus, \( k = 4 \).

### Conclusion
The value of \( k \) is \( 4 \).