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 use the property that two vectors are perpendicular if their dot product is zero.

### Step-by-Step Solution:

1. **Write the expression for the dot product**:
The dot product of two vectors \(\vec{a}\) and \(\vec{b}\) is given by:
\[ \vec{a} \cdot \vec{b} = (5\hat{i} + \lambda \hat{j} + \hat{k}) \cdot (\hat{i} - 2\hat{j} + \hat{k}) \]

2. **Calculate the dot product**:
Using the distributive property of the dot product:
\[ \vec{a} \cdot \vec{b} = 5 \cdot 1 + \lambda \cdot (-2) + 1 \cdot 1 \]
Simplifying this, we get:
\[ \vec{a} \cdot \vec{b} = 5 - 2\lambda + 1 \]
Therefore:
\[ \vec{a} \cdot \vec{b} = 6 - 2\lambda \]

3. **Set the dot product to zero**:
Since the vectors are perpendicular, we set the dot product equal to zero:
\[ 6 - 2\lambda = 0 \]

4. **Solve for \(\lambda\)**:
Rearranging the equation gives:
\[ 2\lambda = 6 \]
Dividing both sides by 2:
\[ \lambda = 3 \]

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