Question

If position vector is given as \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\) and if its signs are reversed, then which of the following physical quantity remains unaffected?

• A. Velocity
• B. Displacement
• C. Acceleration
• D. Torque

Hint:

Quantities involving a {cross product} of two vectors (such as torque or angular momentum) can remain unchanged when both vectors reverse direction.

Answer 1

The Correct Option is D

To solve this problem, we need to understand how reversing the signs of the position vector \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\) affects different physical quantities. When the signs of the position vector's components are reversed, \(\vec{r}\) becomes \(-x\hat{i} - y\hat{j} - z\hat{k}\).

1. Displacement: Displacement is a vector quantity that describes a change in position. Reversing the signs of the position vector affects displacement, as it directly depends on the initial and final position vectors.
2. Velocity: Velocity is the rate of change of displacement with respect to time. Since displacement is affected, velocity also gets affected by the reversal of the position vector's signs.
3. Acceleration: Acceleration is the rate of change of velocity with time. As velocity is affected by the change in position vector, acceleration is also affected.
4. Torque: Torque \(\vec{\tau}\) is given by the cross product of the position vector \(\vec{r}\) and the force \(\vec{F}\): \(\vec{\tau} = \vec{r} \times \vec{F}\). When the position vector \(\vec{r}\) is reversed, torque becomes: \(\vec{\tau} = (-x\hat{i} - y\hat{j} - z\hat{k}) \times \vec{F}\). Since the cross product depends on the perpendicular distance from the axis of rotation, the magnitude of torque remains unaffected by sign reversal.

This analysis shows that reversing the position vector's signs affects displacement, velocity, and acceleration but leaves torque unchanged. Therefore, the correct answer is that the quantity that remains unaffected is Torque.