If `u=ax+by+cz`, `v=ay+bz+cx`, `w=ax+bx+cy`, then the value of `|{:(a,b,c),(b,c,a),(c,a,b):}|xx|{:(x,y,z),(y,z,x),(z,x,y):}|` is

Options

A. `u^(2)+v^(2)+w^(2)-2uvw`
B. `u^(3)+v^(3)+w^(3)-3uvw`
C. `0`
D. none of these

Explanation

To find the value of the determinant \( | \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix} | \times | \begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix} | \), we will follow these steps:

### Step 1: Calculate the first determinant
We need to calculate the determinant \( | \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix} | \).

Using the formula for the determinant of a 3x3 matrix:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where:
- \( a = a, b = b, c = c \)
- \( d = b, e = c, f = a \)
- \( g = c, h = a, i = b \)

Calculating:
\[ D = a(ca - ab) - b(ba - ac) + c(ba - cb) \]
This simplifies to:
\[ D = a(c^2 - ab) - b(b^2 - ac) + c(ab - c^2) \]

### Step 2: Calculate the second determinant
Now we calculate the second determinant \( | \begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix} | \).

Using the same determinant formula:
\[ D' = x(zi - yj) - y(xi - zy) + z(xj - yj) \]
where:
- \( x = x, y = y, z = z \)
- \( d = y, e = z, f = x \)
- \( g = z, h = x, i = y \)

Calculating:
\[ D' = x(z^2 - xy) - y(x^2 - zy) + z(xy - z^2) \]

### Step 3: Combine the determinants
Now we multiply the two determinants:
\[ | \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix} | \times | \begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix} | \]

### Step 4: Substitute values for \( u, v, w \)
From the problem statement, we have:
- \( u = ax + by + cz \)
- \( v = ay + bz + cx \)
- \( w = az + bx + cy \)

### Step 5: Final expression
Using the values of \( u, v, w \) in the determinant, we can express the final result as:
\[ u^3 + v^3 + w^3 - 3uvw = 0 \]

### Conclusion
Thus, the value of the determinant \( | \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix} | \times | \begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix} | \) is given by the expression \( u^3 + v^3 + w^3 - 3uvw \).