What is the value of `lambda` for which
`(lambda hat(i)+hat(j)-hat(k))xx(3hat(i)-2hat(j)+4hat(k))=(2hat(i)-11 hat(j)-7 hat(k))`?

Options

A. 2
B. -2
C. 1
D. 7

Explanation

To solve the problem, we need to find the value of \( \lambda \) such that:

\[ (\lambda \hat{i} + \hat{j} - \hat{k}) \times (3\hat{i} - 2\hat{j} + 4\hat{k}) = 2\hat{i} - 11\hat{j} - 7\hat{k} \]

### Step 1: Set up the cross product in determinant form

We can represent the cross product using a determinant:

\[ \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \lambda & 1 & -1 \\ 3 & -2 & 4 \end{vmatrix} \]

### Step 2: Calculate the determinant

Using the determinant formula for a 3x3 matrix, we calculate:

\[ = \hat{i} \begin{vmatrix} 1 & -1 \\ -2 & 4 \end{vmatrix} - \hat{j} \begin{vmatrix} \lambda & -1 \\ 3 & 4 \end{vmatrix} + \hat{k} \begin{vmatrix} \lambda & 1 \\ 3 & -2 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. For \( \hat{i} \):
\[ 1 \cdot 4 - (-1) \cdot (-2) = 4 - 2 = 2 \]

2. For \( \hat{j} \):
\[ \lambda \cdot 4 - (-1) \cdot 3 = 4\lambda + 3 \]

3. For \( \hat{k} \):
\[ \lambda \cdot (-2) - 1 \cdot 3 = -2\lambda - 3 \]

Putting it all together, we have:

\[ = 2\hat{i} - (4\lambda + 3)\hat{j} + (-2\lambda - 3)\hat{k} \]

### Step 3: Set the result equal to the right-hand side

Now we set the result equal to the right-hand side of the equation:

\[ 2\hat{i} - (4\lambda + 3)\hat{j} + (-2\lambda - 3)\hat{k} = 2\hat{i} - 11\hat{j} - 7\hat{k} \]

### Step 4: Equate coefficients

From the equation, we can equate the coefficients of \( \hat{i}, \hat{j}, \) and \( \hat{k} \):

1. For \( \hat{i} \):
\[ 2 = 2 \quad \text{(This is true)} \]

2. For \( \hat{j} \):
\[ -(4\lambda + 3) = -11 \implies 4\lambda + 3 = 11 \]

3. For \( \hat{k} \):
\[ -2\lambda - 3 = -7 \implies 2\lambda + 3 = 7 \]

### Step 5: Solve the equations

From the \( \hat{j} \) equation:

\[ 4\lambda + 3 = 11 \implies 4\lambda = 8 \implies \lambda = 2 \]

From the \( \hat{k} \) equation:

\[ 2\lambda + 3 = 7 \implies 2\lambda = 4 \implies \lambda = 2 \]

### Conclusion

Thus, the value of \( \lambda \) is:

\[ \lambda = 2 \]