Find the value of :
(i) `(hat(i) xxhat(j))*hat (k) + hat(i)* hat(j)` (ii) `(hat(k) xx hat(j))* hat(i) +hat(j)* hat(k)`
`hat(i) xx (hat(j) + hat(k) )+hat(j) xx(hat(k) +hat(i))+ hat(k) xx (hat(i)+hat(j))`

Explanation

To solve the given problems step by step, we will use the properties of the cross product and dot product of the unit vectors \( \hat{i}, \hat{j}, \hat{k} \).

### (i) Find the value of \( (\hat{i} \times \hat{j}) \cdot \hat{k} + \hat{i} \cdot \hat{j} \)

1. **Calculate \( \hat{i} \times \hat{j} \)**:
- By the right-hand rule and the properties of cross products, we know:
\[ \hat{i} \times \hat{j} = \hat{k} \]

2. **Calculate \( (\hat{i} \times \hat{j}) \cdot \hat{k} \)**:
- Substitute \( \hat{k} \) from the previous step:
\[ \hat{k} \cdot \hat{k} = 1 \]

3. **Calculate \( \hat{i} \cdot \hat{j} \)**:
- The dot product of two orthogonal unit vectors is:
\[ \hat{i} \cdot \hat{j} = 0 \]

4. **Combine the results**:
- Now, add the results from steps 2 and 3:
\[ 1 + 0 = 1 \]

Thus, the value for part (i) is **1**.

### (ii) Find the value of \( (\hat{k} \times \hat{j}) \cdot \hat{i} + \hat{j} \cdot \hat{k} \)

1. **Calculate \( \hat{k} \times \hat{j} \)**:
- Using the right-hand rule:
\[ \hat{k} \times \hat{j} = -\hat{i} \]

2. **Calculate \( (\hat{k} \times \hat{j}) \cdot \hat{i} \)**:
- Substitute \( -\hat{i} \):
\[ -\hat{i} \cdot \hat{i} = -1 \]

3. **Calculate \( \hat{j} \cdot \hat{k} \)**:
- The dot product of two orthogonal unit vectors is:
\[ \hat{j} \cdot \hat{k} = 0 \]

4. **Combine the results**:
- Now, add the results from steps 2 and 3:
\[ -1 + 0 = -1 \]

Thus, the value for part (ii) is **-1**.

### (iii) Find the value of \( \hat{i} \times (\hat{j} + \hat{k}) + \hat{j} \times (\hat{k} + \hat{i}) + \hat{k} \times (\hat{i} + \hat{j}) \)

1. **Calculate \( \hat{i} \times (\hat{j} + \hat{k}) \)**:
- Distributing the cross product:
\[ \hat{i} \times \hat{j} + \hat{i} \times \hat{k} = \hat{k} + (-\hat{j}) = \hat{k} - \hat{j} \]

2. **Calculate \( \hat{j} \times (\hat{k} + \hat{i}) \)**:
- Distributing the cross product:
\[ \hat{j} \times \hat{k} + \hat{j} \times \hat{i} = \hat{i} + (-\hat{k}) = \hat{i} - \hat{k} \]

3. **Calculate \( \hat{k} \times (\hat{i} + \hat{j}) \)**:
- Distributing the cross product:
\[ \hat{k} \times \hat{i} + \hat{k} \times \hat{j} = (-\hat{j}) + (-\hat{i}) = -\hat{j} - \hat{i} \]

4. **Combine all results**:
- Now, add the results from steps 1, 2, and 3:
\[ (\hat{k} - \hat{j}) + (\hat{i} - \hat{k}) + (-\hat{j} - \hat{i}) \]
- Simplifying:
\[ \hat{k} - \hat{j} + \hat{i} - \hat{k} - \hat{j} - \hat{i} = -2\hat{j} \]

Thus, the value for part (iii) is **-2\hat{j}**.

### Summary of Results:
- (i) **1**
- (ii) **-1**
- (iii) **-2\hat{j}**