If `(a times b) times (c times d)=[a b d]c+kd` then the value of k is

Options

A. [b a c]
B. [a b c]
C. [b c d]
D. [c b d]

Answer

To solve the problem `(a × b) × (c × d) = [a b d]c + kd`, we need to find the value of `k`. Let's break this down step by step.

### Step 1: Understand the Cross Product
We start with the expression `(a × b) × (c × d)`. We can use the vector triple product identity, which states that:
\[ x × (y × z) = (x · z)y - (x · y)z \]
In our case, we can let \( x = a \), \( y = b \), and \( z = (c × d) \).

### Step 2: Apply the Vector Triple Product Identity
Using the identity, we can rewrite our expression:
\[ (a × b) × (c × d) = (a · (c × d))b - (b · (c × d))a \]

### Step 3: Simplify the Right Side
We know that the expression on the right side is given as:
\[ [a b d]c + kd \]
Here, \([a b d]\) represents the scalar triple product, which can be computed as the determinant of the matrix formed by the vectors \(a\), \(b\), and \(d\).

### Step 4: Equate the Two Sides
Now we need to equate the two expressions we have:
\[ (a · (c × d))b - (b · (c × d))a = [a b d]c + kd \]

### Step 5: Identify the Coefficients
To find \(k\), we need to analyze the coefficients of \(d\) on both sides of the equation.
- The term \(kd\) on the right side suggests that we need to isolate the terms involving \(d\) from the left side.

### Step 6: Analyze the Left Side
From the left side, we need to express the terms in a way that isolates \(d\):
- The term \(a · (c × d)\) will yield a scalar that multiplies \(b\).
- The term \(- (b · (c × d))a\) will yield a scalar that multiplies \(a\).

### Step 7: Recognize the Scalar Triple Product
The expression \((c × d)\) can be rewritten using the properties of determinants:
\[ (c × d) = [c d a]a + [c d b]b \]
This means we can express the left side in terms of the scalar triple products.

### Step 8: Solve for k
After equating and simplifying, we will find that:
\[ k = -[a b d] \]
This means that the value of \(k\) is the negative of the scalar triple product of \(a\), \(b\), and \(d\).

### Final Answer
Thus, the value of \(k\) is:
\[ k = -[a b d] \]