If `(a xx b) xx c = a xx (bxx c)`, then

Options

A. `b xx( c xx a) =0`
B. `( c xx a) xx b =0`
C. `c xx( a xxb) =0`
D. none of these

Answer

To solve the equation `(a × b) × c = a × (b × c)`, we will use the properties of the vector triple product. The vector triple product identity states that:

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

This identity allows us to rewrite the left-hand side and right-hand side of the equation.

### Step 1: Rewrite the Left-Hand Side
Using the vector triple product identity, we can rewrite the left-hand side:

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

### Step 2: Rewrite the Right-Hand Side
Now, we apply the vector triple product identity to the right-hand side:

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

### Step 3: Set the Two Sides Equal
Now we have:

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

### Step 4: Rearranging the Equation
Rearranging the equation gives us:

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

### Step 5: Grouping Like Terms
We can group the terms involving \(a\) and \(b\):

\[ (c \cdot b)a + (c \cdot a)b = (a \cdot b)c + (a \cdot c)b \]

### Step 6: Analyzing the Result
For the equality to hold, the coefficients of \(a\) and \(b\) on both sides must be equal. This leads us to conclude that:

\[ (c \cdot b) = (a \cdot b) \quad \text{and} \quad (c \cdot a) = (a \cdot c) \]

### Conclusion
Thus, we have shown that the equation holds true under the vector triple product identity, confirming that:

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