The pair of vector quantities haivng the same dimensions is

Options

A. Force and Impulse
B. Moment of inertia and Moment of the couple
C. Surface tension and Force constant
D. Thrust andd weight

Answer

To solve the problem of identifying the pair of vector quantities having the same dimensions, we will analyze the given options step by step.

### Step 1: Identify the dimensions of each vector quantity

1. **Force (F)**:
- Formula: \( F = m \cdot a \) (mass times acceleration)
- Dimensions: \( [F] = [M][L][T^{-2}] \)

2. **Impulse (I)**:
- Formula: \( I = \Delta p = m \cdot v \) (change in momentum)
- Dimensions: \( [I] = [M][L][T^{-1}] \)

3. **Moment of Inertia (I)**:
- Formula: \( I = m \cdot r^2 \) (mass times radius squared)
- Dimensions: \( [I] = [M][L^2] \)

4. **Moment of Couple (C)**:
- Formula: \( C = F \cdot r \) (force times distance)
- Dimensions: \( [C] = [M][L][T^{-2}][L] = [M][L^2][T^{-2}] \)

5. **Thrust (T)**:
- Formula: \( T = F \) (force acting perpendicularly)
- Dimensions: \( [T] = [M][L][T^{-2}] \)

6. **Weight (W)**:
- Formula: \( W = m \cdot g \) (mass times acceleration due to gravity)
- Dimensions: \( [W] = [M][L][T^{-2}] \)

### Step 2: Compare the dimensions of the vector quantities

- **Force and Impulse**:
- Dimensions: \( [F] = [M][L][T^{-2}] \) and \( [I] = [M][L][T^{-1}] \) (not the same)

- **Moment of Inertia and Moment of Couple**:
- Dimensions: \( [I] = [M][L^2] \) and \( [C] = [M][L^2][T^{-2}] \) (not the same)

- **Thrust and Weight**:
- Dimensions: \( [T] = [M][L][T^{-2}] \) and \( [W] = [M][L][T^{-2}] \) (same dimensions)

### Conclusion

The pair of vector quantities having the same dimensions is **Thrust and Weight**.

---