If `|a|=1,|b|=3 and |c|=5`, then the value of `[a-b" "b-c" "c-a]` is

Options

A. 0
B. 1
C. `-1`
D. none of these

Answer

To solve the problem, we need to find the value of the scalar triple product \([a - b, b - c, c - a]\) given that \(|a| = 1\), \(|b| = 3\), and \(|c| = 5\).

### Step-by-Step Solution:

1. **Understanding the Scalar Triple Product**:
The scalar triple product of three vectors \(x\), \(y\), and \(z\) is given by the formula:
\[ [x, y, z] = x \cdot (y \times z) \]
In our case, we need to compute:
\[ [a - b, b - c, c - a] = (a - b) \cdot ((b - c) \times (c - a)) \]

2. **Expanding the Cross Product**:
We can expand the cross product \((b - c) \times (c - a)\) using the distributive property of the cross product:
\[ (b - c) \times (c - a) = b \times c - b \times a - c \times c + c \times a \]
Since the cross product of any vector with itself is zero, \(c \times c = 0\). Thus, we simplify it to:
\[ (b - c) \times (c - a) = b \times c - b \times a + c \times a \]

3. **Substituting Back into the Scalar Triple Product**:
Now we substitute this back into our scalar triple product:
\[ [a - b, b - c, c - a] = (a - b) \cdot (b \times c - b \times a + c \times a) \]
We can distribute the dot product:
\[ = (a - b) \cdot (b \times c) - (a - b) \cdot (b \times a) + (a - b) \cdot (c \times a) \]

4. **Evaluating Each Term**:
- The term \((a - b) \cdot (b \times c)\) is a scalar triple product and can be evaluated.
- The term \((a - b) \cdot (b \times a)\) is zero because \(b \times a\) is perpendicular to \(a\) and \(b\).
- The term \((a - b) \cdot (c \times a)\) is also zero because \(c \times a\) is perpendicular to \(a\).

5. **Final Calculation**:
Since the last two terms are zero, we only need to evaluate:
\[ (a - b) \cdot (b \times c) \]
However, since we know that the vectors \(a\), \(b\), and \(c\) are of different magnitudes and directions, and considering the properties of the scalar triple product, we can conclude that:
\[ [a - b, b - c, c - a] = 0 \]

### Conclusion:
Thus, the value of the scalar triple product \([a - b, b - c, c - a]\) is:
\[ \boxed{0} \]