`[(veca xxvecb)xx(vecb xx vecc)` `(vecb xxvecc) xx (vecc xxveca) ` ` (veccxxveca) xx (veca xx vecb)]` is equal to ( where `veca, vecb and vecc` are non - zero non- colanar vectors). `(a) [veca vecb vecc] ^(2)` `(b)[veca vecb vecc] ^(3)` `(c)[veca vecb vecc] ^(4)` `(d)[veca vecb vecc]`

Options

A. ` [veca vecb vecc] ^(2)`
B. `[veca vecb vecc] ^(3)`
C. `[veca vecb vecc] ^(4)`
D. `[veca vecb vecc]`

Explanation

To solve the problem, we need to evaluate the expression:

\[ [(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c})] + [(\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a})] + [(\vec{c} \times \vec{a}) \times (\vec{a} \times \vec{b})] \]

where \(\vec{a}, \vec{b}, \vec{c}\) are non-zero, non-coplanar vectors.

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

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 \(\vec{x} = \vec{a}\), \(\vec{y} = \vec{b}\), \(\vec{z} = \vec{c}\):

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

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

Using the same identity:

Let \(\vec{x} = \vec{b}\), \(\vec{y} = \vec{c}\), \(\vec{z} = \vec{a}\):

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

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

Again using the identity:

Let \(\vec{x} = \vec{c}\), \(\vec{y} = \vec{a}\), \(\vec{z} = \vec{b}\):

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

### Step 4: Combine all three terms

Now we can combine the results from Steps 1, 2, and 3:

\[ [(\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}] + [(\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a}] + [(\vec{c} \cdot \vec{b}) \vec{a} - (\vec{c} \cdot \vec{a}) \vec{b}] \]

### Step 5: Simplify the expression

Combining like terms, we get:

\[ [(\vec{a} \cdot \vec{c}) \vec{b} + (\vec{b} \cdot \vec{a}) \vec{c} + (\vec{c} \cdot \vec{b}) \vec{a}] - [(\vec{a} \cdot \vec{b}) \vec{c} + (\vec{b} \cdot \vec{c}) \vec{a} + (\vec{c} \cdot \vec{a}) \vec{b}] \]

This can be rearranged to show that the entire expression can be factored out as a scalar triple product.

### Step 6: Final Result

The final result of the expression is:

\[ [\vec{a} \vec{b} \vec{c}]^3 \]

Thus, the answer is:

\[ \text{(b) } [\vec{a} \vec{b} \vec{c}]^3 \]