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 triple product \([a - b, b - c, c - a]\) given that \(|a| = 1\), \(|b| = 3\), and \(|c| = 5\).

### Step-by-Step Solution:

1. **Understanding the Triple Product**:
The triple product \([u, v, w]\) of three vectors \(u\), \(v\), and \(w\) is defined as the scalar volume of the parallelepiped formed by these vectors. It can be calculated using the determinant of a matrix formed by these vectors.

2. **Expressing the Vectors**:
We have three vectors:
- \(u = a - b\)
- \(v = b - c\)
- \(w = c - a\)

3. **Adding the Vectors**:
Let's add these vectors:
\[ u + v + w = (a - b) + (b - c) + (c - a) = 0 \]
This shows that the vectors \(u\), \(v\), and \(w\) are linearly dependent.

4. **Linear Dependence and Coplanarity**:
Since \(u + v + w = 0\), the vectors \(u\), \(v\), and \(w\) are coplanar. When three vectors are coplanar, their triple product is zero.

5. **Conclusion**:
Therefore, the value of the triple product \([a - b, b - c, c - a]\) is:
\[ [a - b, b - c, c - a] = 0 \]

### Final Answer:
The value of \([a - b, b - c, c - a]\) is **0**.

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