`a xx [ a xx (a xx b)]` equals

Options

A. `(a.a) (axxb)`
B. `(a.a)(bxxa)`
C. `(b.b)(axxb)`
D. `(b.b)(bxxa)`

Answer

To solve the expression \( a \times [ a \times (a \times b)] \), we will use the vector triple product identity. The vector triple product identity states that:

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

### Step-by-Step Solution:

1. **Identify the Inner Product**:
We start with the innermost expression \( a \times b \). Let’s denote this as \( c \):
\[ c = a \times b \]

2. **Apply the Triple Product Identity**:
Now we need to evaluate \( a \times (a \times c) \). According to the vector triple product identity:
\[ a \times (a \times c) = (a \cdot c)a - (a \cdot a)c \]
Substituting \( c = a \times b \):
\[ a \times (a \times (a \times b)) = (a \cdot (a \times b))a - (a \cdot a)(a \times b) \]

3. **Evaluate the Dot Products**:
- The term \( a \cdot (a \times b) \) is equal to zero because the dot product of a vector with a vector that is perpendicular to it (which is the case here) is zero:
\[ a \cdot (a \times b) = 0 \]
- The term \( a \cdot a \) is simply the magnitude squared of vector \( a \):
\[ a \cdot a = |a|^2 \]

4. **Substituting Back**:
Now substituting these results back into our expression:
\[ a \times (a \times (a \times b)) = 0 \cdot a - |a|^2(a \times b) \]
This simplifies to:
\[ a \times (a \times (a \times b)) = -|a|^2(a \times b) \]

5. **Final Result**:
Therefore, the final result of the expression \( a \times [ a \times (a \times b)] \) is:
\[ -|a|^2(a \times b) \]