Let x, y and z be unit vectors such that
`abs(x-y)^(2)+abs(y-z)^(2)+abs(z-x)^(2)=9`
Then `abs(x+y-z)^(2)-4x.y=`

Options

A. 1
B. 4
C. 6
D. 8

Answer

To solve the problem, we need to evaluate the expression \( |x + y - z|^2 - 4x \cdot y \) given that \( |x - y|^2 + |y - z|^2 + |z - x|^2 = 9 \), where \( x, y, z \) are unit vectors.

### Step 1: Expand the given equation
We start with the equation:
\[ |x - y|^2 + |y - z|^2 + |z - x|^2 = 9 \]
Using the formula \( |a - b|^2 = |a|^2 + |b|^2 - 2a \cdot b \), we can expand each term:
\[ |x - y|^2 = |x|^2 + |y|^2 - 2x \cdot y \]
\[ |y - z|^2 = |y|^2 + |z|^2 - 2y \cdot z \]
\[ |z - x|^2 = |z|^2 + |x|^2 - 2z \cdot x \]
Since \( x, y, z \) are unit vectors, we have \( |x|^2 = |y|^2 = |z|^2 = 1 \). Thus, we can substitute:
\[ |x - y|^2 = 1 + 1 - 2x \cdot y = 2 - 2x \cdot y \]
\[ |y - z|^2 = 1 + 1 - 2y \cdot z = 2 - 2y \cdot z \]
\[ |z - x|^2 = 1 + 1 - 2z \cdot x = 2 - 2z \cdot x \]
Now substituting these into the original equation:
\[ (2 - 2x \cdot y) + (2 - 2y \cdot z) + (2 - 2z \cdot x) = 9 \]
This simplifies to:
\[ 6 - 2(x \cdot y + y \cdot z + z \cdot x) = 9 \]
Rearranging gives:
\[ -2(x \cdot y + y \cdot z + z \cdot x) = 3 \]
Thus,
\[ x \cdot y + y \cdot z + z \cdot x = -\frac{3}{2} \]

### Step 2: Evaluate the expression \( |x + y - z|^2 - 4x \cdot y \)
Next, we need to evaluate:
\[ |x + y - z|^2 - 4x \cdot y \]
Using the expansion formula again:
\[ |x + y - z|^2 = |x|^2 + |y|^2 + |z|^2 + 2(x \cdot y - x \cdot z - y \cdot z) \]
Substituting \( |x|^2 = |y|^2 = |z|^2 = 1 \):
\[ |x + y - z|^2 = 1 + 1 + 1 + 2(x \cdot y - x \cdot z - y \cdot z) = 3 + 2(x \cdot y - x \cdot z - y \cdot z) \]

### Step 3: Substitute the known values
We know from the previous step:
\[ x \cdot y + y \cdot z + z \cdot x = -\frac{3}{2} \]
Let \( A = x \cdot y \), \( B = y \cdot z \), and \( C = z \cdot x \). Then:
\[ A + B + C = -\frac{3}{2} \]
Now, we can express \( B + C \) as:
\[ B + C = -\frac{3}{2} - A \]
Substituting this into our expression:
\[ |x + y - z|^2 = 3 + 2(A - (-\frac{3}{2} - A)) = 3 + 2(A + \frac{3}{2} + A) = 3 + 2(2A + \frac{3}{2}) = 3 + 4A + 3 = 6 + 4A \]
Thus, we have:
\[ |x + y - z|^2 - 4x \cdot y = (6 + 4A) - 4A = 6 \]

### Final Result
The final answer is:
\[ \boxed{6} \]