Find:
(i) "north cross west" `" "` (ii) "down dot south"
(iii) "west dot west" (iv) "south cross south".
Let each vector have unit magnitude.

Answer

To solve the question, we will analyze each part step by step, using vector notation for the directions given. We will denote the unit vectors as follows:

- North = \( \hat{j} \)
- South = \( -\hat{j} \)
- East = \( \hat{i} \)
- West = \( -\hat{i} \)
- Up = \( \hat{k} \)
- Down = \( -\hat{k} \)

Now, let's solve each part of the question.

### Part (i): North cross West

1. **Identify the vectors**: North is represented by \( \hat{j} \) and West is represented by \( -\hat{i} \).
2. **Cross product**: We need to calculate \( \hat{j} \times (-\hat{i}) \).
3. **Using the right-hand rule**: The cross product of two vectors can be determined using the right-hand rule. Point your fingers in the direction of the first vector (North), curl them towards the second vector (West), and your thumb will point in the direction of the resultant vector.
4. **Calculate**:
\[ \hat{j} \times (-\hat{i}) = -(\hat{j} \times \hat{i}) = -(-\hat{k}) = \hat{k} \]
5. **Result**: The result is \( \hat{k} \), which points upwards.

### Part (ii): Down dot South

1. **Identify the vectors**: Down is represented by \( -\hat{k} \) and South is represented by \( -\hat{j} \).
2. **Dot product**: We need to calculate \( (-\hat{k}) \cdot (-\hat{j}) \).
3. **Calculate**: The dot product of two perpendicular unit vectors is zero.
\[ (-\hat{k}) \cdot (-\hat{j}) = 0 \]
4. **Result**: The result is \( 0 \).

### Part (iii): West dot West

1. **Identify the vectors**: West is represented by \( -\hat{i} \).
2. **Dot product**: We need to calculate \( (-\hat{i}) \cdot (-\hat{i}) \).
3. **Calculate**: The dot product of a vector with itself is the square of its magnitude (which is 1 for unit vectors).
\[ (-\hat{i}) \cdot (-\hat{i}) = 1 \]
4. **Result**: The result is \( 1 \).

### Part (iv): South cross South

1. **Identify the vectors**: South is represented by \( -\hat{j} \).
2. **Cross product**: We need to calculate \( (-\hat{j}) \times (-\hat{j}) \).
3. **Calculate**: The cross product of any vector with itself is zero.
\[ (-\hat{j}) \times (-\hat{j}) = 0 \]
4. **Result**: The result is the null vector \( \vec{0} \).

### Summary of Results:
1. North cross West = \( \hat{k} \)
2. Down dot South = \( 0 \)
3. West dot West = \( 1 \)
4. South cross South = \( \vec{0} \)