A scalar quantity and a vector quantity haivng the same dimensions are

Options

A. Gravitational Potential and latent heat
B. Force and Tension
C. Specific heat and gas constant
D. Frequency and velocity gradient

Answer

To solve the question regarding a scalar quantity and a vector quantity having the same dimensions, we will analyze the given options step by step.

### Step 1: Understand Scalar and Vector Quantities
- **Scalar Quantity**: A quantity that has only magnitude and no direction. Examples include mass, temperature, and speed.
- **Vector Quantity**: A quantity that has both magnitude and direction. Examples include force, velocity, and acceleration.

### Step 2: Analyze the Options
We need to identify which pair consists of a scalar and a vector quantity that share the same dimensions.

1. **Option A**: Both quantities are scalar.
2. **Option B**: Both quantities are vector.
3. **Option C**: One is scalar, and the other is vector.
4. **Option D**: One is scalar (frequency) and the other is vector (velocity gradient).

### Step 3: Determine the Dimensions
- **Frequency**:
- Defined as the number of occurrences of a repeating event per unit time.
- Dimension of frequency = \( [T^{-1}] \) (time inverse).

- **Velocity Gradient**:
- Defined as the change in velocity per unit distance.
- If velocity is measured in meters per second (m/s) and distance in meters (m), then:
- Dimension of velocity gradient = \( \frac{[L/T]}{[L]} = [T^{-1}] \) (time inverse).

### Step 4: Compare the Dimensions
Both frequency and velocity gradient have the same dimension of \( [T^{-1}] \).

### Conclusion
Thus, the correct answer is **Option D**, where frequency is a scalar quantity and velocity gradient is a vector quantity, both having the same dimension \( [T^{-1}] \).

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