Let `veca,vecb` and `vecc` be the three non-coplanar vectors and `vecd` be a non zero vector which is perpendicular to `veca+vecb+vecc` and is represented as `vecd=x(vecaxxvecb)+y(vecbxxvecc)+z(veccxxveca)`. Then,

Options

A. `x^(3)+y^(3)+z^(3)=3xyz`
B. `xy+yz+zx=0`
C. `x=y=z`
D. `x^(2)+y^(2)+z^(2)=xy+yz+zx`

Explanation

To solve the problem, we need to analyze the given information about the vectors and their relationships. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We are given three non-coplanar vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), and a vector \(\vec{d}\) that is perpendicular to the sum of these vectors, i.e., \(\vec{d} \perp (\vec{a} + \vec{b} + \vec{c})\). The vector \(\vec{d}\) is expressed as:
\[ \vec{d} = x (\vec{a} \times \vec{b}) + y (\vec{b} \times \vec{c}) + z (\vec{c} \times \vec{a}) \]

### Step 2: Set Up the Perpendicular Condition
Since \(\vec{d}\) is perpendicular to \(\vec{a} + \vec{b} + \vec{c}\), we can write the dot product:
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot \vec{d} = 0 \]

### Step 3: Substitute \(\vec{d}\)
Substituting the expression for \(\vec{d}\) into the dot product gives:
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot \left( x (\vec{a} \times \vec{b}) + y (\vec{b} \times \vec{c}) + z (\vec{c} \times \vec{a}) \right) = 0 \]

### Step 4: Expand the Dot Product
Expanding this dot product, we have:
\[ x (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} \times \vec{b}) + y (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{b} \times \vec{c}) + z (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{c} \times \vec{a}) = 0 \]

### Step 5: Analyze Each Term
Using the property of the scalar triple product, we know that:
- \((\vec{u} \cdot (\vec{v} \times \vec{w})) = \text{Volume of the parallelepiped formed by } \vec{u}, \vec{v}, \vec{w}\)

Thus, each term can be evaluated:
1. \((\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} \times \vec{b}) = 0\) (since \(\vec{a} \times \vec{b}\) is perpendicular to both \(\vec{a}\) and \(\vec{b}\))
2. \((\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{b} \times \vec{c}) = 0\)
3. \((\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{c} \times \vec{a}) = 0\)

### Step 6: Combine the Results
Since all terms equal zero, we can conclude:
\[ x + y + z = 0 \]

### Step 7: Conclusion
The final result leads us to the conclusion that:
\[ x + y + z = 0 \]