Find the value of `lambda` for which `vec(a)` and `vec(b)` are perpendicular, where
(i) `vec(a)=2hat(i)+lambda hat(j)+hat(k)` and `vec(b)=(hat(i)-2hat(j)+3hat(k))`
(ii) `vec(a)=3hat(i)-hat(j)+4hat(k)` and `vec(b)=- lamnda hat(i)+3 hat(j)+3 hat(k)`
(iii) `vec(A)=2hat(i)+4hat(j)-hat(k)` and `vec(b)=3 hat(i)-2 hat(j)+lambda hat(k)`
(iv) `vec(a)=3 hat(i)+2 hat(j)-5 hat(k)` and `vec(b)=-5 hat(j)+lambda hat(k)`

Answer

To find the value of `lambda` for which the vectors `vec(a)` and `vec(b)` are perpendicular, we need to use the property that two vectors are perpendicular if their dot product equals zero.

Let's solve each part step by step.

### Part (i)
Given:
- `vec(a) = 2hat(i) + lambda hat(j) + hat(k)`
- `vec(b) = hat(i) - 2hat(j) + 3hat(k)`

1. **Calculate the dot product**:
\[ vec(a) \cdot vec(b) = (2hat(i) + lambda hat(j) + hat(k)) \cdot (hat(i) - 2hat(j) + 3hat(k)) \]
\[ = 2(1) + lambda(-2) + 1(3) \]
\[ = 2 - 2lambda + 3 \]
\[ = 5 - 2lambda \]

2. **Set the dot product to zero**:
\[ 5 - 2lambda = 0 \]

3. **Solve for `lambda`**:
\[ 2lambda = 5 \implies lambda = \frac{5}{2} \]

### Part (ii)
Given:
- `vec(a) = 3hat(i) - hat(j) + 4hat(k)`
- `vec(b) = -lambda hat(i) + 3hat(j) + 3hat(k)`

1. **Calculate the dot product**:
\[ vec(a) \cdot vec(b) = (3hat(i) - hat(j) + 4hat(k)) \cdot (-lambda hat(i) + 3hat(j) + 3hat(k)) \]
\[ = 3(-lambda) + (-1)(3) + 4(3) \]
\[ = -3lambda - 3 + 12 \]
\[ = -3lambda + 9 \]

2. **Set the dot product to zero**:
\[ -3lambda + 9 = 0 \]

3. **Solve for `lambda`**:
\[ -3lambda = -9 \implies lambda = 3 \]

### Part (iii)
Given:
- `vec(a) = 2hat(i) + 4hat(j) - hat(k)`
- `vec(b) = 3hat(i) - 2hat(j) + lambda hat(k)`

1. **Calculate the dot product**:
\[ vec(a) \cdot vec(b) = (2hat(i) + 4hat(j) - hat(k)) \cdot (3hat(i) - 2hat(j) + lambda hat(k)) \]
\[ = 2(3) + 4(-2) + (-1)(lambda) \]
\[ = 6 - 8 - lambda \]
\[ = -2 - lambda \]

2. **Set the dot product to zero**:
\[ -2 - lambda = 0 \]

3. **Solve for `lambda`**:
\[ lambda = -2 \]

### Part (iv)
Given:
- `vec(a) = 3hat(i) + 2hat(j) - 5hat(k)`
- `vec(b) = -5hat(j) + lambda hat(k)`

1. **Calculate the dot product**:
\[ vec(a) \cdot vec(b) = (3hat(i) + 2hat(j) - 5hat(k)) \cdot (0hat(i) - 5hat(j) + lambda hat(k)) \]
\[ = 3(0) + 2(-5) + (-5)(lambda) \]
\[ = 0 - 10 - 5lambda \]
\[ = -10 - 5lambda \]

2. **Set the dot product to zero**:
\[ -10 - 5lambda = 0 \]

3. **Solve for `lambda`**:
\[ -5lambda = 10 \implies lambda = -2 \]

### Summary of Results
1. For Part (i), \( \lambda = \frac{5}{2} \)
2. For Part (ii), \( \lambda = 3 \)
3. For Part (iii), \( \lambda = -2 \)
4. For Part (iv), \( \lambda = -2 \)