The value of `lambda` for which the two vectors `vec(a) = 5hat(i) + lambda hat(j) + hat(k)` and `vec(b) = hat(i) - 2hat(j) + hat(k)` are perpendicular to each other is :

Options

A. 2
B. `-2`
C. 3
D. `-3`

Answer

To find the value of \( \lambda \) for which the vectors \( \vec{a} = 5\hat{i} + \lambda\hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} - 2\hat{j} + \hat{k} \) are perpendicular, we can follow these steps:

### Step 1: Understand the condition for perpendicular vectors
Two vectors are perpendicular if their dot product is equal to zero. Therefore, we need to find \( \vec{a} \cdot \vec{b} = 0 \).

### Step 2: Write down the dot product
The dot product of the vectors \( \vec{a} \) and \( \vec{b} \) is given by:
\[ \vec{a} \cdot \vec{b} = (5\hat{i} + \lambda\hat{j} + \hat{k}) \cdot (\hat{i} - 2\hat{j} + \hat{k}) \]

### Step 3: Calculate the dot product
Using the properties of dot products:
\[ \vec{a} \cdot \vec{b} = 5 \cdot 1 + \lambda \cdot (-2) + 1 \cdot 1 \]
This simplifies to:
\[ 5 - 2\lambda + 1 = 0 \]

### Step 4: Simplify the equation
Combine like terms:
\[ 6 - 2\lambda = 0 \]

### Step 5: Solve for \( \lambda \)
Rearranging the equation gives:
\[ 2\lambda = 6 \]
Dividing both sides by 2:
\[ \lambda = 3 \]

### Conclusion
The value of \( \lambda \) for which the two vectors are perpendicular is \( \lambda = 3 \).

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