`[((vecaxxvecb)xx(vecbxxvecc),(vecbxxvecc)xx(veccxxveca),(veccxxveca)xx(vecaxxvecb))]` equal to

Options

A. `[(veca,vecb,vecc)]^(2)`
B. `[(veca,vecb,vecc)]^(3)`
C. `[(veca,vecb,vecc)]^(4)`
D. none of these

Explanation

To solve the problem, we need to evaluate the expression given, which involves the scalar triple products of three vectors. The expression is:

\[ \left[ (\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}), (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}), (\vec{c} \times \vec{a}) \times (\vec{a} \times \vec{b}) \right] \]

We will evaluate each term step-by-step using the vector triple product identity.

### Step 1: Evaluate \((\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c})\)

Using the vector triple product identity:

\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]

Let \(\vec{x} = \vec{a} \times \vec{b}\), \(\vec{y} = \vec{b}\), and \(\vec{z} = \vec{c}\).

Thus,

\[ (\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}) = \left((\vec{a} \times \vec{b}) \cdot \vec{c}\right) \vec{b} - \left((\vec{a} \times \vec{b}) \cdot \vec{b}\right) \vec{c} \]

Since \((\vec{a} \times \vec{b}) \cdot \vec{b} = 0\) (the dot product of a vector with itself is zero), we have:

\[ (\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}) = \left((\vec{a} \times \vec{b}) \cdot \vec{c}\right) \vec{b} \]

### Step 2: Evaluate \((\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a})\)

Using the same identity:

Let \(\vec{x} = \vec{b} \times \vec{c}\), \(\vec{y} = \vec{c}\), and \(\vec{z} = \vec{a}\).

Thus,

\[ (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}) = \left((\vec{b} \times \vec{c}) \cdot \vec{a}\right) \vec{c} - \left((\vec{b} \times \vec{c}) \cdot \vec{c}\right) \vec{a} \]

Again, \((\vec{b} \times \vec{c}) \cdot \vec{c} = 0\), so we have:

\[ (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}) = \left((\vec{b} \times \vec{c}) \cdot \vec{a}\right) \vec{c} \]

### Step 3: Evaluate \((\vec{c} \times \vec{a}) \times (\vec{a} \times \vec{b})\)

Using the identity again:

Let \(\vec{x} = \vec{c} \times \vec{a}\), \(\vec{y} = \vec{a}\), and \(\vec{z} = \vec{b}\).

Thus,

\[ (\vec{c} \times \vec{a}) \times (\vec{a} \times \vec{b}) = \left((\vec{c} \times \vec{a}) \cdot \vec{b}\right) \vec{a} - \left((\vec{c} \times \vec{a}) \cdot \vec{a}\right) \vec{b} \]

Again, \((\vec{c} \times \vec{a}) \cdot \vec{a} = 0\), so we have:

\[ (\vec{c} \times \vec{a}) \times (\vec{a} \times \vec{b}) = \left((\vec{c} \times \vec{a}) \cdot \vec{b}\right) \vec{a} \]

### Final Expression

Now, we can combine the results from all three steps:

\[ \left[ \left((\vec{a} \times \vec{b}) \cdot \vec{c}\right) \vec{b}, \left((\vec{b} \times \vec{c}) \cdot \vec{a}\right) \vec{c}, \left((\vec{c} \times \vec{a}) \cdot \vec{b}\right) \vec{a} \right] \]

### Conclusion

The final result is a vector whose components depend on the scalar triple products of the vectors involved.