If the vectors `a = i-j+2k, b =2i+4j+k` and `c=lambdai+j+mu k` are mutually orthogonal, then `(lambda,mu) = `

Options

A. `(-3,2)`
B. `(2,-3)`
C. `(-2,3)`
D. `(3,-2)`

Answer

To solve the problem of finding the values of \((\lambda, \mu)\) such that the vectors \( \mathbf{a} = \mathbf{i} - \mathbf{j} + 2\mathbf{k} \), \( \mathbf{b} = 2\mathbf{i} + 4\mathbf{j} + \mathbf{k} \), and \( \mathbf{c} = \lambda \mathbf{i} + \mathbf{j} + \mu \mathbf{k} \) are mutually orthogonal, we will follow these steps:

### Step 1: Set up the dot product conditions
Since the vectors are mutually orthogonal, the dot product of each pair of vectors must equal zero:
1. \( \mathbf{a} \cdot \mathbf{c} = 0 \)
2. \( \mathbf{b} \cdot \mathbf{c} = 0 \)

### Step 2: Calculate \( \mathbf{a} \cdot \mathbf{c} \)
The dot product \( \mathbf{a} \cdot \mathbf{c} \) is calculated as follows:
\[ \mathbf{a} \cdot \mathbf{c} = (1)(\lambda) + (-1)(1) + (2)(\mu) = \lambda - 1 + 2\mu \]
Setting this equal to zero gives us:
\[ \lambda + 2\mu - 1 = 0 \quad \text{(Equation 1)} \]

### Step 3: Calculate \( \mathbf{b} \cdot \mathbf{c} \)
Next, we calculate \( \mathbf{b} \cdot \mathbf{c} \):
\[ \mathbf{b} \cdot \mathbf{c} = (2)(\lambda) + (4)(1) + (1)(\mu) = 2\lambda + 4 + \mu \]
Setting this equal to zero gives us:
\[ 2\lambda + \mu + 4 = 0 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
We now have a system of two equations:
1. \( \lambda + 2\mu = 1 \) (from Equation 1)
2. \( 2\lambda + \mu = -4 \) (from Equation 2)

To solve this system, we can multiply Equation 1 by 2:
\[ 2\lambda + 4\mu = 2 \quad \text{(Equation 3)} \]

Now we can subtract Equation 2 from Equation 3:
\[ (2\lambda + 4\mu) - (2\lambda + \mu) = 2 - (-4) \]
This simplifies to:
\[ 3\mu = 6 \]
Thus, we find:
\[ \mu = 2 \]

### Step 5: Substitute back to find \( \lambda \)
Now we substitute \( \mu = 2 \) back into Equation 1:
\[ \lambda + 2(2) = 1 \]
This simplifies to:
\[ \lambda + 4 = 1 \implies \lambda = 1 - 4 = -3 \]

### Final Answer
Thus, the values of \( (\lambda, \mu) \) are:
\[ (\lambda, \mu) = (-3, 2) \]