Suppose a, b, c are three vectors such that `abs(a)=abs(b)=abs(c)=1" and "a+b+c=0," then "abs(a-b)^(2)+abs(b-c)^(2)+abs(c-a)^(2)=`_______

Answer

To solve the problem, we will follow these steps:

Given:
- \( |a| = |b| = |c| = 1 \)
- \( a + b + c = 0 \)

We need to find:
\[ |a - b|^2 + |b - c|^2 + |c - a|^2 \]

### Step 1: Rewrite the expression using the dot product
We can express the squared magnitudes in terms of dot products:
\[ |a - b|^2 = (a - b) \cdot (a - b) = a \cdot a - 2a \cdot b + b \cdot b \]
Since \( |a|^2 = 1 \) and \( |b|^2 = 1 \), we have:
\[ |a - b|^2 = 1 - 2a \cdot b + 1 = 2 - 2a \cdot b \]
Thus,
\[ |a - b|^2 + |b - c|^2 + |c - a|^2 = (2 - 2a \cdot b) + (2 - 2b \cdot c) + (2 - 2c \cdot a) \]

### Step 2: Combine the terms
Combining the terms gives:
\[ |a - b|^2 + |b - c|^2 + |c - a|^2 = 6 - 2(a \cdot b + b \cdot c + c \cdot a) \]

### Step 3: Find \( a \cdot b + b \cdot c + c \cdot a \)
Using the condition \( a + b + c = 0 \), we can take the dot product with \( a \):
\[ a \cdot (a + b + c) = 0 \implies a \cdot a + a \cdot b + a \cdot c = 0 \]
This simplifies to:
\[ 1 + a \cdot b + a \cdot c = 0 \implies a \cdot b + a \cdot c = -1 \]

Now, taking the dot product with \( b \):
\[ b \cdot (a + b + c) = 0 \implies b \cdot a + b \cdot b + b \cdot c = 0 \]
This simplifies to:
\[ b \cdot a + 1 + b \cdot c = 0 \implies b \cdot a + b \cdot c = -1 \]

Finally, taking the dot product with \( c \):
\[ c \cdot (a + b + c) = 0 \implies c \cdot a + c \cdot b + c \cdot c = 0 \]
This simplifies to:
\[ c \cdot a + c \cdot b + 1 = 0 \implies c \cdot a + c \cdot b = -1 \]

Now we have:
1. \( a \cdot b + a \cdot c = -1 \)
2. \( b \cdot a + b \cdot c = -1 \)
3. \( c \cdot a + c \cdot b = -1 \)

Adding these three equations:
\[ (a \cdot b + a \cdot c) + (b \cdot a + b \cdot c) + (c \cdot a + c \cdot b) = -3 \]
This simplifies to:
\[ 2(a \cdot b + b \cdot c + c \cdot a) = -3 \implies a \cdot b + b \cdot c + c \cdot a = -\frac{3}{2} \]

### Step 4: Substitute back into the expression
Now substituting back into our expression:
\[ |a - b|^2 + |b - c|^2 + |c - a|^2 = 6 - 2\left(-\frac{3}{2}\right) = 6 + 3 = 9 \]

### Final Answer
Thus, the value of \( |a - b|^2 + |b - c|^2 + |c - a|^2 \) is:
\[ \boxed{9} \]