If `( a times b)*(c times d)=(a*c)(b*d)+k(a.d)(b.c)` then the value of k is

Options

A. 1
B. 0
C. -2
D. -1

Answer

To solve the problem, we need to analyze the given equation:

\[ ( \mathbf{a} \times \mathbf{b} ) \times ( \mathbf{c} \times \mathbf{d} ) = (\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d}) + k(\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c}) \]

We will use the vector triple product identity to simplify the left-hand side of the equation.

### Step 1: Apply the Vector Triple Product Identity
The vector triple product identity states that:

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

Using this identity, we can rewrite the left-hand side:

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

### Step 2: Expand the Left-Hand Side
Now, we will expand the left-hand side using the identity we just applied:

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

### Step 3: Compare Both Sides
Now we need to compare the expanded left-hand side with the right-hand side of the original equation:

\[ (\mathbf{a} \cdot \mathbf{d}) \mathbf{c} - (\mathbf{a} \cdot \mathbf{c}) \mathbf{d} = (\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d}) + k(\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c}) \]

### Step 4: Rearranging Terms
Rearranging the terms gives us:

\[ (\mathbf{a} \cdot \mathbf{d}) \mathbf{c} - (\mathbf{a} \cdot \mathbf{c}) \mathbf{d} = (\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d}) + k(\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c}) \]

### Step 5: Isolate k
To find \( k \), we can isolate the terms involving \( k \):

Comparing coefficients of \( \mathbf{a} \cdot \mathbf{d} \) and \( \mathbf{a} \cdot \mathbf{c} \):

1. Coefficient of \( \mathbf{c} \) on the left side is \( \mathbf{a} \cdot \mathbf{d} \)
2. Coefficient of \( \mathbf{d} \) on the left side is \( -(\mathbf{a} \cdot \mathbf{c}) \)

From the right side, we have:

- Coefficient of \( \mathbf{c} \) is \( k(\mathbf{b} \cdot \mathbf{d}) \)
- Coefficient of \( \mathbf{d} \) is \( (\mathbf{a} \cdot \mathbf{c}) \)

Setting these equal gives us:

\[ k = -1 \]

### Final Answer
Thus, the value of \( k \) is:

\[ \boxed{-1} \]

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