Find the value of `lamda` for which `3hat(i) + 4hat(j) - hat(k) and -2 hat(i) + lamda hat(j) + 10 hat(k)` are perpendicular?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the value of \( \lambda \) for which the vectors \( \mathbf{A} = 3\hat{i} + 4\hat{j} - \hat{k} \) and \( \mathbf{B} = -2\hat{i} + \lambda\hat{j} + 10\hat{k} \) are perpendicular, we can follow these steps:

### Step 1: Set up the dot product condition
Two vectors are perpendicular if their dot product is zero. Therefore, we need to calculate the dot product of \( \mathbf{A} \) and \( \mathbf{B} \) and set it equal to zero.

\[ \mathbf{A} \cdot \mathbf{B} = (3\hat{i} + 4\hat{j} - \hat{k}) \cdot (-2\hat{i} + \lambda\hat{j} + 10\hat{k}) = 0 \]

### Step 2: Calculate the dot product
Using the properties of the dot product, we calculate:

\[ \mathbf{A} \cdot \mathbf{B} = 3(-2) + 4(\lambda) + (-1)(10) \]

Calculating each term:
- \( 3 \cdot -2 = -6 \)
- \( 4 \cdot \lambda = 4\lambda \)
- \( -1 \cdot 10 = -10 \)

Putting it all together, we have:

\[ -6 + 4\lambda - 10 = 0 \]

### Step 3: Simplify the equation
Combine the constant terms:

\[ 4\lambda - 16 = 0 \]

### Step 4: Solve for \( \lambda \)
Now, isolate \( \lambda \):

\[ 4\lambda = 16 \]

Dividing both sides by 4 gives:

\[ \lambda = 4 \]

### Final Answer
The value of \( \lambda \) for which the vectors are perpendicular is \( \lambda = 4 \).

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