Find Torque Calculator Calc 3

Calculate Torque (τ = r x F)

Enter the components of the position vector (r) and the force vector (F) to calculate the torque vector (τ) and its magnitude using the cross product method often taught in Calc 3.

Position Vector (r)

Force Vector (F)

Input vectors (r, F) and resulting torque vector (τ) with magnitudes.

Vector	x-comp	y-comp	z-comp	Magnitude
r (m)	1.00	2.00	0.00	2.24
F (N)	0.00	10.00	0.00	10.00
τ (N·m)	0.00	0.00	10.00	10.00

What is Torque (from 3D Vectors – Calc 3 Method)?

Torque, often called a moment of force, is a measure of the rotational force applied to an object. When dealing with forces and positions in three-dimensional space, as is common in physics and engineering (and topics covered in Calculus 3 or "Calc 3"), torque is calculated as a vector quantity using the cross product of the position vector (r) and the force vector (F). The Torque Calculator (from 3D Vectors – Calc 3 Method) implements this vector cross product.

The position vector r goes from the axis of rotation (or pivot point) to the point where the force F is applied. The resulting torque vector τ is perpendicular to both r and F, and its direction is given by the right-hand rule. The magnitude of the torque represents the turning effect.

This Torque Calculator (from 3D Vectors – Calc 3 Method) is useful for students learning vector calculus (like in Calc 3), engineers designing rotating systems, and physicists analyzing forces and motion.

Common misconceptions include thinking torque is the same as force (it's force *at a distance* causing rotation) or that it's always just force times lever arm length (which is only true for the magnitude when the force is perpendicular to the position vector in 2D).

Torque Formula and Mathematical Explanation (Vector Cross Product)

The torque vector τ is defined as the cross product of the position vector r and the force vector F:

τ = r x F

If we have r = (rx, ry, rz) and F = (Fx, Fy, Fz) in Cartesian coordinates, the cross product is calculated as:

τx = (ry * Fz) – (rz * Fy)

τy = (rz * Fx) – (rx * Fz)

τz = (rx * Fy) – (ry * Fx)

So, the torque vector is τ = (τx, τy, τz).

The magnitude of the torque vector is given by:

|τ| = √(τx² + τy² + τz²)

Alternatively, the magnitude can be expressed as |τ| = |r| |F| sin(θ), where θ is the angle between r and F. Our Torque Calculator (from 3D Vectors – Calc 3 Method) uses the component-wise calculation.

Variables Table

Variable	Meaning	Unit	Typical Range
rx, ry, rz	Components of the position vector r	meters (m)	-1000 to 1000
Fx, Fy, Fz	Components of the force vector F	Newtons (N)	-10000 to 10000
τx, τy, τz	Components of the torque vector τ	Newton-meters (N·m)	Depends on inputs
|τ|	Magnitude of the torque vector	Newton-meters (N·m)	0 to large values

Practical Examples (Real-World Use Cases)

Understanding how to use the Torque Calculator (from 3D Vectors – Calc 3 Method) is best illustrated with examples.

Example 1: Tightening a Bolt

Imagine tightening a bolt with a wrench. The bolt is at the origin (0,0,0). You apply a force at the end of the wrench, say at position r = (0.3, 0.1, 0) m, and the force is F = (0, 50, 0) N (straight up along y).

• rx=0.3, ry=0.1, rz=0
• Fx=0, Fy=50, Fz=0

Using the Torque Calculator (from 3D Vectors – Calc 3 Method):

• τx = (0.1*0) – (0*50) = 0 N·m
• τy = (0*0) – (0.3*0) = 0 N·m
• τz = (0.3*50) – (0.1*0) = 15 N·m
• |τ| = 15 N·m

The torque is 15 N·m around the z-axis, causing the bolt to rotate.

Example 2: Force on a Door

You push a door at a point r = (0, 0.8, 0.1) m relative to the hinges, with a force F = (10, 0, 0) N (pushing horizontally along x).

• rx=0, ry=0.8, rz=0.1
• Fx=10, Fy=0, Fz=0

Using the Torque Calculator (from 3D Vectors – Calc 3 Method):

• τx = (0.8*0) – (0.1*0) = 0 N·m
• τy = (0.1*10) – (0*0) = 1 N·m
• τz = (0*0) – (0.8*10) = -8 N·m
• |τ| = √(0² + 1² + (-8)²) = √65 ≈ 8.06 N·m

The torque has components along y and z, causing the door to rotate mainly around the z-axis (hinges).

How to Use This Torque Calculator (from 3D Vectors – Calc 3 Method)

1. Enter Position Vector (r): Input the x, y, and z components (rx, ry, rz) of the position vector from the axis/pivot to where the force is applied, in meters.
2. Enter Force Vector (F): Input the x, y, and z components (Fx, Fy, Fz) of the force vector, in Newtons.
3. Calculate: The calculator automatically updates the torque vector components (τx, τy, τz) and the torque magnitude |τ| in Newton-meters (N·m) as you type. You can also click "Calculate Torque".
4. Read Results: The primary result is the magnitude |τ|. The intermediate results show the components τx, τy, and τz. The table and chart also visualize these values.
5. Reset: Click "Reset" to return to default values.
6. Copy: Click "Copy Results" to copy the main results and inputs to your clipboard.

The Torque Calculator (from 3D Vectors – Calc 3 Method) provides instant feedback, helping you understand the relationship between force, position, and torque in 3D.

Key Factors That Affect Torque Results

• Magnitude of Position Vector (|r|): A larger distance from the pivot to the point of force application generally increases torque for a given force and angle.
• Magnitude of Force Vector (|F|): A larger force magnitude results in a larger torque for a given position vector and angle.
• Direction of Position Vector (r): The components of r determine how the force components contribute to the torque components.
• Direction of Force Vector (F): Similarly, the direction of F is crucial. Torque is maximized when F is perpendicular to r.
• Angle Between r and F: The sine of the angle between r and F directly influences the torque magnitude. A 90-degree angle maximizes torque; 0 or 180 degrees results in zero torque.
• Coordinate System: The values of the components depend on the chosen coordinate system and origin (pivot point).

Our Torque Calculator (from 3D Vectors – Calc 3 Method) correctly uses these vector components to find the torque.

Frequently Asked Questions (FAQ)

What is the unit of torque?

The SI unit of torque is the Newton-meter (N·m).

Is torque a vector or a scalar?

Torque is a vector quantity, having both magnitude and direction. This Torque Calculator (from 3D Vectors – Calc 3 Method) calculates the vector components and the magnitude.

What is the 'right-hand rule' for torque?

The right-hand rule helps determine the direction of the torque vector τ = r x F. Point your fingers in the direction of r, then curl them towards F. Your thumb will point in the direction of τ.

Can torque be negative?

Yes, the components of the torque vector (τx, τy, τz) can be negative, indicating direction along the negative axes. The magnitude is always non-negative.

What if the force is applied at the pivot point?

If the force is applied at the pivot, the position vector r is zero, and thus the torque τ is zero, regardless of the force.

How does this relate to Calc 3?

In Calculus 3 (Multivariable/Vector Calculus), the cross product of vectors is a key topic, and calculating torque as r x F is a standard application. This Torque Calculator (from 3D Vectors – Calc 3 Method) directly applies that principle.

What's the difference between torque and work?

Torque is a rotational force (N·m), while work is energy transferred (Joules, which is also N·m dimensionally, but used differently). Torque causes angular acceleration; work is done when torque causes rotation over an angle.

Can I use 2D vectors with this calculator?

Yes, for 2D problems in the x-y plane, simply set rz = 0 and Fz = 0. The torque will then only have a z-component (τz).

Related Tools and Internal Resources

• Vector Addition Calculator: Calculate the sum of two or more vectors.
• Dot Product Calculator: Find the dot product of two vectors.
• Kinematics Calculator: Analyze motion with constant acceleration.
• Cross Product Calculator: Specifically calculate the cross product of two 3D vectors. This is the core of our Torque Calculator (from 3D Vectors – Calc 3 Method).
• Moment of Inertia Calculator: Calculate the rotational inertia of various shapes.
• Gear Ratio Calculator: Understand how gears transmit torque and speed.