In vector form, torque τ on a particle is defined as τ = r × F, where r is the position vector and F is the applied force. In which direction does the torque vector point relative to r and F?

Introduction / Context:
Torque (or moment of force) is a vector quantity that measures the turning effect of a force about a point or axis. In three dimensional mechanics, torque is defined using a vector cross product, τ = r × F, where r is the position vector from the axis to the point of application of the force, and F is the force vector. This question asks you to identify the direction of the torque vector relative to r and F.

Given Data / Assumptions:

• The torque vector τ is defined by the cross product τ = r × F.
• r and F lie in some plane through the origin or pivot point.
• We use the standard right-hand rule convention for cross products.
• We are interested in the direction of τ, not its magnitude.

Concept / Approach:
The cross product of two vectors a × b is a vector that is perpendicular to the plane containing a and b. Its direction is given by the right-hand rule: if you point the fingers of your right hand along a and curl them toward b, your thumb points in the direction of a × b. Thus, for τ = r × F, the torque vector is perpendicular to both r and F and points along the axis about which rotation tends to occur. It is not parallel to the force, nor opposite to it, and is not generally parallel to the position vector r.

Step-by-Step Solution:
Step 1: Write the vector definition τ = r × F. Step 2: Recall the geometric rule for a cross product: the result is perpendicular to the plane containing the two vectors. Step 3: Visualise r and F lying in a plane; the torque vector must then come out of or go into this plane. Step 4: Apply the right-hand rule: align the fingers of your right hand with r and curl them toward F. Step 5: The thumb then points in the direction of τ, perpendicular to both r and F. Step 6: Conclude that the correct description is that τ is perpendicular to the plane containing r and F, given by the right-hand rule.

Verification / Alternative check:
In rotational dynamics around a fixed axis, we often treat torque as pointing along the axis of rotation. For example, when you turn a screw, the force is applied tangentially, the radius extends from the axis to the point of application and the torque vector points along the screw axis. This is consistent with the cross product direction being perpendicular to the plane of r and F. Problems in textbooks that use right-handed coordinate systems also show torque vectors along axes, further confirming this relationship.

Why Other Options Are Wrong:
Parallel to the applied force F: If torque were parallel to F, it would not correctly represent the sense of rotation relative to the axis defined by r and F; the cross product definition contradicts this.

Exactly opposite to F: Torque is not defined as a negative of the force vector; its direction is determined by the perpendicular to the plane of r and F.

Always parallel to r: In many simple problems r and F are perpendicular, but τ is still perpendicular to both; it does not align with r except in special degenerate cases.

Common Pitfalls:
Students sometimes confuse the direction of torque with that of force because both are involved in rotational motion. Another confusion is between the scalar magnitude of torque r * F * sin(θ) and the vector direction. To avoid these mistakes, always remember that the cross product provides both magnitude and a direction perpendicular to the plane of the vectors, and the right-hand rule is the standard tool to find that direction.

Final Answer:
The torque vector points perpendicular to the plane containing r and F, as given by the right-hand rule.