Vector A is pointing eastwards and vector B northwards .If `|A|=|B|` then match the following two columns.
`{:(,"Column I ",,"Column II"),((A),(A+B),(P),"north -east"),((B),(A-B),(Q),"Vertically upwards"),((C), (AxxB),(r),"south-east"),((D), (AxxB)xx(AxxB),(s),"None"):}`

Explanation

To solve the problem step by step, we will analyze the vectors A and B, their operations, and match the results with the options provided.

### Step 1: Define the Vectors
- Vector A is pointing eastward, which we can represent as \( \vec{A} = |A| \hat{i} \).
- Vector B is pointing northward, which we can represent as \( \vec{B} = |B| \hat{j} \).
- Given that \( |\vec{A}| = |\vec{B}| \), we can denote their magnitudes as \( |A| = |B| = k \).

### Step 2: Calculate \( \vec{A} + \vec{B} \)
- The sum of the two vectors is:
\[ \vec{A} + \vec{B} = |A| \hat{i} + |B| \hat{j} = k \hat{i} + k \hat{j} \]
- This resultant vector is at a 45-degree angle to both the east and north directions, pointing towards the north-east.
- **Match**: This corresponds to option P ("north-east").

### Step 3: Calculate \( \vec{A} - \vec{B} \)
- The difference of the two vectors is:
\[ \vec{A} - \vec{B} = |A| \hat{i} - |B| \hat{j} = k \hat{i} - k \hat{j} \]
- This resultant vector is also at a 45-degree angle, but it points towards the south-east direction.
- **Match**: This corresponds to option R ("south-east").

### Step 4: Calculate \( \vec{A} \times \vec{B} \)
- The cross product of the two vectors is:
\[ \vec{A} \times \vec{B} = |A| |B| (\hat{i} \times \hat{j}) = k^2 \hat{k} \]
- The direction of this vector is vertically upwards (along the z-axis).
- **Match**: This corresponds to option Q ("vertically upwards").

### Step 5: Calculate \( (\vec{A} \times \vec{B}) \times (\vec{A} \times \vec{B}) \)
- Since \( \vec{A} \times \vec{B} = k^2 \hat{k} \), we have:
\[ (\vec{A} \times \vec{B}) \times (\vec{A} \times \vec{B}) = (k^2 \hat{k}) \times (k^2 \hat{k}) = 0 \]
- The cross product of any vector with itself is zero, which has no direction.
- **Match**: This corresponds to option S ("None").

### Final Matches
- A: \( \vec{A} + \vec{B} \) → P ("north-east")
- B: \( \vec{A} - \vec{B} \) → R ("south-east")
- C: \( \vec{A} \times \vec{B} \) → Q ("vertically upwards")
- D: \( (\vec{A} \times \vec{B}) \times (\vec{A} \times \vec{B}) \) → S ("None")