If `vecd=vecaxxvecb+vecbxxvecc+veccxxveca` is a on zero vector and `|(vecd.vecc)(vecaxxvecb)+(vecd.veca)(vecbxxvecc)+(vecd.vecb)(veccxxveca)|=0` then (A) `|veca|+|vecb|+|vecc|=|vecd|` (B) `|veca|=|vecb|=|vecc|` (C) `veca,vecb,vecc` are coplanar (D) `veca+vecc=vec(2b)`

Options

A. `|veca|=|vecb|=|vecc|`
B. `|veca|+|vecb|+|vecc|=|vecd|`
C. `veca, vecb and vecc` are coplanar
D. none of these

Explanation

To solve the problem, we need to analyze the given information step by step.

### Step 1: Understanding the Given Expression
We start with the expression for the vector **d**:
\[ \vec{d} = \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} \]
We know that **d** is a non-zero vector.

### Step 2: Analyzing the Dot Products
Next, we need to analyze the dot products given in the problem:
\[ |\vec{d} \cdot \vec{c} (\vec{a} \times \vec{b}) + \vec{d} \cdot \vec{a} (\vec{b} \times \vec{c}) + \vec{d} \cdot \vec{b} (\vec{c} \times \vec{a})| = 0 \]
This implies that the entire expression evaluates to zero.

### Step 3: Simplifying the Dot Products
We can simplify each dot product:
- \(\vec{d} \cdot \vec{a} = \vec{a} \times \vec{b} \cdot \vec{a} + \vec{b} \times \vec{c} \cdot \vec{a} + \vec{c} \times \vec{a} \cdot \vec{a} = 0 + \vec{b} \times \vec{c} \cdot \vec{a} + 0 = \vec{b} \times \vec{c} \cdot \vec{a}\)
- \(\vec{d} \cdot \vec{b} = \vec{a} \times \vec{b} \cdot \vec{b} + \vec{b} \times \vec{c} \cdot \vec{b} + \vec{c} \times \vec{a} \cdot \vec{b} = 0 + 0 + \vec{c} \times \vec{a} \cdot \vec{b} = \vec{c} \times \vec{a} \cdot \vec{b}\)
- \(\vec{d} \cdot \vec{c} = \vec{a} \times \vec{b} \cdot \vec{c} + \vec{b} \times \vec{c} \cdot \vec{c} + \vec{c} \times \vec{a} \cdot \vec{c} = \vec{a} \times \vec{b} \cdot \vec{c} + 0 + 0 = \vec{a} \times \vec{b} \cdot \vec{c}\)

### Step 4: Substituting Back into the Expression
Substituting these results back into the expression gives us:
\[ |\vec{a} \times \vec{b} \cdot \vec{c} (\vec{a} \times \vec{b}) + \vec{b} \times \vec{c} \cdot \vec{a} (\vec{b} \times \vec{c}) + \vec{c} \times \vec{a} \cdot \vec{b} (\vec{c} \times \vec{a})| = 0 \]

### Step 5: Factoring Out Common Terms
We can factor out the common term \(\vec{a} \times \vec{b} \cdot \vec{c}\):
\[ |\vec{a} \times \vec{b} \cdot \vec{c} \cdot (\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a})| = 0 \]
Since \(\vec{d}\) is a non-zero vector, the term \(\vec{a} \times \vec{b} \cdot \vec{c}\) must equal zero.

### Step 6: Conclusion
The condition \(\vec{a} \times \vec{b} \cdot \vec{c} = 0\) implies that the vectors \(\vec{a}, \vec{b}, \vec{c}\) are coplanar.

Thus, the correct option is:
**(C) \(\vec{a}, \vec{b}, \vec{c}\) are coplanar.**