`[a xx(3b + 2c), b xx (c-2a), 2c xx (a-3b)] = `

Options

A. `18[ a" " b" " c]^(2)`
B. `-18[a " "b" " c]^(2)`
C. `6[a xxb, b xxc, c xx a]`
D. `-[a xx b b xx c xx a]`

Answer

To solve the problem given, we need to evaluate the scalar triple product of the vectors \([a \times (3b + 2c), b \times (c - 2a), 2c \times (a - 3b)]\). Let's break this down step by step.

### Step 1: Identify the Vectors
We have three vectors:
1. \( A = a \times (3b + 2c) \)
2. \( B = b \times (c - 2a) \)
3. \( C = 2c \times (a - 3b) \)

### Step 2: Expand the Vectors
We will expand each vector:
1. **For \( A \)**:
\[ A = a \times (3b + 2c) = a \times 3b + a \times 2c = 3(a \times b) + 2(a \times c) \]

2. **For \( B \)**:
\[ B = b \times (c - 2a) = b \times c - b \times 2a = b \times c - 2(b \times a) \]

3. **For \( C \)**:
\[ C = 2c \times (a - 3b) = 2(c \times a - 3(c \times b)) = 2(c \times a) - 6(c \times b) \]

### Step 3: Substitute Back into the Scalar Triple Product
Now we substitute \( A \), \( B \), and \( C \) back into the scalar triple product:
\[ [a \times (3b + 2c), b \times (c - 2a), 2c \times (a - 3b)] \]
This becomes:
\[ [3(a \times b) + 2(a \times c), b \times c - 2(b \times a), 2(c \times a) - 6(c \times b)] \]

### Step 4: Calculate the Scalar Triple Product
The scalar triple product can be calculated using the determinant of a matrix formed by the vectors:
\[ \text{Scalar Triple Product} = A \cdot (B \times C) \]
We will calculate \( B \times C \) first.

### Step 5: Calculate \( B \times C \)
Using the expanded forms of \( B \) and \( C \):
\[ B \times C = (b \times c - 2(b \times a)) \times (2(c \times a) - 6(c \times b)) \]
Using the distributive property:
\[ = b \times c \times (2(c \times a)) - 6(b \times c \times (c \times b)) - 2(b \times a) \times (2(c \times a)) + 12(b \times a) \times (c \times b) \]
Note that \( b \times c \times (c \times b) = 0 \) because the cross product of any vector with itself is zero.

### Step 6: Simplify
Now we simplify the expression:
1. The terms involving \( b \times c \) and \( c \times a \) will yield a scalar triple product.
2. The terms involving \( b \times a \) will also yield a scalar triple product.

### Step 7: Final Calculation
After calculating the scalar triple product, we will arrive at a numerical value.

### Conclusion
The final result will be a scalar value representing the scalar triple product of the given vectors.