The value of `[a times b, b times c, c times a]` is

Options

A. 2 [a b c]
B. [a b c]
C. `[a b c]^(2)`
D. 0

Answer

To solve the problem of finding the value of the scalar triple product \([a \times b, b \times c, c \times a]\), we will follow these steps:

### Step 1: Understand the Scalar Triple Product
The scalar triple product of three vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) is defined as:
\[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \]
This represents the volume of the parallelepiped formed by the three vectors and is a scalar quantity.

### Step 2: Identify the Vectors
In our case, we have three vectors:
- \( \mathbf{p} = \mathbf{a} \times \mathbf{b} \)
- \( \mathbf{q} = \mathbf{b} \times \mathbf{c} \)
- \( \mathbf{r} = \mathbf{c} \times \mathbf{a} \)

We need to find the scalar triple product of these vectors:
\[ [\mathbf{p}, \mathbf{q}, \mathbf{r}] = \mathbf{p} \cdot (\mathbf{q} \times \mathbf{r}) \]

### Step 3: Calculate \( \mathbf{q} \times \mathbf{r} \)
Using the vector triple product identity:
\[ \mathbf{q} \times \mathbf{r} = (\mathbf{b} \times \mathbf{c}) \times (\mathbf{c} \times \mathbf{a}) \]
This can be simplified using the vector triple product formula:
\[ \mathbf{x} \times (\mathbf{y} \times \mathbf{z}) = (\mathbf{x} \cdot \mathbf{z}) \mathbf{y} - (\mathbf{x} \cdot \mathbf{y}) \mathbf{z} \]
Let \( \mathbf{x} = \mathbf{b} \), \( \mathbf{y} = \mathbf{c} \), and \( \mathbf{z} = \mathbf{a} \):
\[ \mathbf{q} \times \mathbf{r} = (\mathbf{b} \cdot \mathbf{a}) \mathbf{c} - (\mathbf{b} \cdot \mathbf{c}) \mathbf{a} \]

### Step 4: Calculate \( \mathbf{p} \cdot (\mathbf{q} \times \mathbf{r}) \)
Now we substitute \( \mathbf{p} \):
\[ \mathbf{p} = \mathbf{a} \times \mathbf{b} \]
Thus,
\[ [\mathbf{p}, \mathbf{q}, \mathbf{r}] = (\mathbf{a} \times \mathbf{b}) \cdot \left[(\mathbf{b} \cdot \mathbf{a}) \mathbf{c} - (\mathbf{b} \cdot \mathbf{c}) \mathbf{a}\right] \]

### Step 5: Expand the Dot Product
Using the distributive property of the dot product:
\[ = (\mathbf{a} \times \mathbf{b}) \cdot \left[(\mathbf{b} \cdot \mathbf{a}) \mathbf{c}\right] - (\mathbf{b} \cdot \mathbf{c})(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{a} \]

### Step 6: Evaluate Each Term
1. The first term:
- \( (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} = [\mathbf{a}, \mathbf{b}, \mathbf{c}] \)
- This is the scalar triple product and represents the volume.

2. The second term:
- \( (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{a} = 0 \) (since the cross product is orthogonal to both vectors).

### Step 7: Final Result
Thus, we conclude that:
\[ [\mathbf{p}, \mathbf{q}, \mathbf{r}] = [\mathbf{a}, \mathbf{b}, \mathbf{c}] \]
The value of \([a \times b, b \times c, c \times a]\) is the scalar triple product of vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \).

### Conclusion
The value of \([a \times b, b \times c, c \times a]\) is:
\[ \boxed{[\mathbf{a}, \mathbf{b}, \mathbf{c}]} \]