x,y,z are distinct scalars such that `[xa+yb+zc, xb+yc+za, xc+ya+zb] =0` where a,b,c are non-coplanar vectors, then

Options

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

Answer

To solve the problem, we start with the given equation:

\[ [xa + yb + zc, xb + yc + za, xc + ya + zb] = 0 \]

where \( a, b, c \) are non-coplanar vectors and \( x, y, z \) are distinct scalars.

### Step 1: Understand the Scalar Triple Product
The expression given is a scalar triple product. The scalar triple product of three vectors \( u, v, w \) is given by:

\[ [u, v, w] = u \cdot (v \times w) \]

In our case, we can denote:

- \( u = xa + yb + zc \)
- \( v = xb + yc + za \)
- \( w = xc + ya + zb \)

### Step 2: Set Up the Scalar Triple Product
We can express the scalar triple product as:

\[ [xa + yb + zc, xb + yc + za, xc + ya + zb] = 0 \]

This means that the vectors \( u, v, w \) are coplanar.

### Step 3: Expand the Scalar Triple Product
Using the properties of the scalar triple product, we can expand the expression:

\[ [xa + yb + zc, xb + yc + za, xc + ya + zb] = (xa + yb + zc) \cdot ((xb + yc + za) \times (xc + ya + zb)) \]

### Step 4: Calculate the Cross Product
Now, we need to compute the cross product \( (xb + yc + za) \times (xc + ya + zb) \). Using the distributive property of the cross product:

\[ (xb + yc + za) \times (xc + ya + zb) = xb \times xc + xb \times ya + xb \times zb + yc \times xc + yc \times ya + yc \times zb + za \times xc + za \times ya + za \times zb \]

### Step 5: Substitute and Simplify
Since \( a, b, c \) are non-coplanar, we can use the fact that the scalar triple product is zero if and only if the vectors are coplanar. Thus, we need to analyze the coefficients of the resulting expression.

### Step 6: Set Up the Equation
From the expansion, we can derive the coefficients for each vector. We will find that:

\[ x^3 + y^3 + z^3 - 3xyz = 0 \]

This is a well-known identity that holds true when \( x+y+z = 0 \) or when \( x, y, z \) are equal.

### Step 7: Analyze the Conditions
Since \( x, y, z \) are distinct scalars, the only solution that satisfies the equation \( x^3 + y^3 + z^3 - 3xyz = 0 \) is:

\[ x + y + z = 0 \]

### Conclusion
Thus, the final result is:

\[ \boxed{x + y + z = 0} \]