Find `'lambda'` and `'mu'` if :
`(hat(i)+3hat(j)+9hat(k))xx(3hat(i)-lambda hat(j)+mu hat(k))=hat(0)`.

Options

A. `lambda =-9` and `mu = 27`
B. `lambda =9` and `mu = 27`
C. `lambda =-3` and `mu = 27`
D. `lambda =-9` and `mu = -27`

Answer

To solve the problem, we need to find the values of \( \lambda \) and \( \mu \) such that the cross product of the vectors \( \hat{i} + 3\hat{j} + 9\hat{k} \) and \( 3\hat{i} - \lambda\hat{j} + \mu\hat{k} \) equals the zero vector \( \hat{0} \).

### Step-by-Step Solution:

1. **Set up the cross product**:
We need to compute the cross product:
\[ (\hat{i} + 3\hat{j} + 9\hat{k}) \times (3\hat{i} - \lambda\hat{j} + \mu\hat{k}) = \hat{0} \]

2. **Write the determinant**:
The cross product can be expressed as a determinant:
\[ \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 3 & 9 \\ 3 & -\lambda & \mu \end{vmatrix} \]

3. **Expand the determinant**:
We will expand this determinant using the first row:
\[ = \hat{i} \begin{vmatrix} 3 & 9 \\ -\lambda & \mu \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 9 \\ 3 & \mu \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 3 \\ 3 & -\lambda \end{vmatrix} \]

4. **Calculate the 2x2 determinants**:
- For \( \hat{i} \):
\[ \begin{vmatrix} 3 & 9 \\ -\lambda & \mu \end{vmatrix} = 3\mu + 9\lambda \]
- For \( \hat{j} \):
\[ \begin{vmatrix} 1 & 9 \\ 3 & \mu \end{vmatrix} = 1 \cdot \mu - 9 \cdot 3 = \mu - 27 \]
- For \( \hat{k} \):
\[ \begin{vmatrix} 1 & 3 \\ 3 & -\lambda \end{vmatrix} = 1 \cdot (-\lambda) - 3 \cdot 3 = -\lambda - 9 \]

5. **Combine the results**:
Putting it all together, we have:
\[ (3\mu + 9\lambda)\hat{i} - (\mu - 27)\hat{j} + (-\lambda - 9)\hat{k} = \hat{0} \]

6. **Set coefficients equal to zero**:
For the equation to hold true, the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) must all equal zero:
\[ 3\mu + 9\lambda = 0 \quad (1) \]
\[ \mu - 27 = 0 \quad (2) \]
\[ -\lambda - 9 = 0 \quad (3) \]

7. **Solve the equations**:
From equation (2):
\[ \mu = 27 \]
From equation (3):
\[ -\lambda = 9 \implies \lambda = -9 \]

8. **Final values**:
Thus, the values of \( \lambda \) and \( \mu \) are:
\[ \lambda = -9, \quad \mu = 27 \]