Find The Work Done By The Force Vector Calculator

Calculate Work Done

Enter the components of the force vector (in Newtons) and the displacement vector (in meters) to find the work done.

Input Vectors Summary

Vector	X-comp	Y-comp	Z-comp	Magnitude
Force (F)	10	5	2	11.36
Displacement (D)	3	4	1	5.10

Table showing the components and magnitudes of the input force and displacement vectors.

Vector Components Visualization

Bar chart visualizing the magnitudes of the x, y, and z components of the force and displacement vectors.

What is the Work Done by a Force Vector Calculator?

The Work Done by a Force Vector Calculator is a tool used to determine the amount of work performed by a constant force when it acts on an object, causing it to undergo a certain displacement. When both the force and displacement are represented as vectors, the work done is calculated as the dot product (or scalar product) of these two vectors.

This calculator is particularly useful in physics and engineering to understand how energy is transferred to or from an object via the action of a force along a displacement. It simplifies the calculation when force and displacement are given in component form (e.g., Fx, Fy, Fz and Dx, Dy, Dz).

Who should use it?

• Physics students learning about work, energy, and power.
• Engineers analyzing forces and motion in mechanical systems.
• Scientists and researchers dealing with vector quantities.
• Anyone needing to calculate the scalar product of two 3D vectors representing force and displacement.

Common Misconceptions

• Work is always force times distance: This is only true if the force and displacement are in the same direction and the force is constant. The Work Done by a Force Vector Calculator correctly uses the dot product, which accounts for the angle between the force and displacement vectors.
• Work is a vector: Work is a scalar quantity, meaning it has magnitude but no direction. It is the result of the dot product of two vectors.
• Only the force in the direction of motion does work: More precisely, only the component of the force along the direction of displacement does work. The dot product inherently calculates this.

Work Done by a Force Vector Calculator Formula and Mathematical Explanation

When a constant force vector F acts on an object, causing a displacement vector d, the work done (W) by the force is defined as the dot product (or scalar product) of F and d:

W = F ⋅ d

If the force vector is given by F = Fx i + Fy j + Fz k and the displacement vector is d = Dx i + Dy j + Dz k, where i, j, and k are the unit vectors along the x, y, and z axes respectively, then the work done is:

W = (Fx * Dx) + (Fy * Dy) + (Fz * Dz)

Alternatively, the work done can also be expressed in terms of the magnitudes of the force (|F|), the magnitude of the displacement (|d|), and the angle (θ) between the force and displacement vectors:

W = |F| |d| cos(θ)

where:

• |F| = sqrt(Fx² + Fy² + Fz²) is the magnitude of the force.
• |d| = sqrt(Dx² + Dy² + Dz²) is the magnitude of the displacement.
• θ is the angle between F and d, which can be found using cos(θ) = (F ⋅ d) / (|F| |d|).

Variables Table

Variable	Meaning	Unit	Typical Range
Fx, Fy, Fz	Components of the Force vector	Newtons (N)	Any real number
Dx, Dy, Dz	Components of the Displacement vector	Meters (m)	Any real number
|F|	Magnitude of the Force vector	Newtons (N)	≥ 0
|D|	Magnitude of the Displacement vector	Meters (m)	≥ 0
W	Work Done	Joules (J)	Any real number (can be negative)
θ	Angle between Force and Displacement vectors	Degrees (°) or Radians (rad)	0° to 180° (0 to π rad)

Variables used in the Work Done by a Force Vector Calculator.

Practical Examples (Real-World Use Cases)

Example 1: Pushing a Box

Imagine you are pushing a box on the floor. You apply a force F = (20 i + 10 j + 0 k) N, and the box moves by a displacement d = (5 i + 2 j + 0 k) m.

• Fx = 20 N, Fy = 10 N, Fz = 0 N
• Dx = 5 m, Dy = 2 m, Dz = 0 m

Work Done (W) = (20 * 5) + (10 * 2) + (0 * 0) = 100 + 20 + 0 = 120 Joules.

The work done by your force is 120 J.

Example 2: Force at an Angle

A force F = (5 i – 3 j + 2 k) N acts on an object, causing a displacement d = (2 i + 4 j – 1 k) m.

• Fx = 5 N, Fy = -3 N, Fz = 2 N
• Dx = 2 m, Dy = 4 m, Dz = -1 m

Work Done (W) = (5 * 2) + (-3 * 4) + (2 * -1) = 10 – 12 – 2 = -4 Joules.

In this case, the work done is negative, which means the force (or at least its dominant component relative to displacement) is generally opposing the displacement, or the object is losing energy due to this force over this displacement.

How to Use This Work Done by a Force Vector Calculator

1. Enter Force Components: Input the x, y, and z components of the force vector (Fx, Fy, Fz) in Newtons (N) into the respective fields.
2. Enter Displacement Components: Input the x, y, and z components of the displacement vector (Dx, Dy, Dz) in meters (m) into their fields.
3. View Results: The calculator will instantly update and display:
   • The total Work Done (in Joules).
   • The magnitude of the force vector.
   • The magnitude of the displacement vector.
   • The angle between the force and displacement vectors (if magnitudes are non-zero).
4. Interpret the Results: Positive work means the force contributes to increasing the object's kinetic energy (if it's the net force), negative work means it reduces it, and zero work means the force is perpendicular to the displacement or one of them is zero.
5. Reset: Click "Reset" to clear the fields and start with default values.
6. Copy: Click "Copy Results" to copy the main results and inputs to your clipboard.

Our dot product calculator can also be used for similar vector calculations.

Key Factors That Affect Work Done Results

1. Force Components (Fx, Fy, Fz): The magnitude and direction of the force vector directly influence the work done. Larger force components generally lead to larger work, depending on displacement.
2. Displacement Components (Dx, Dy, Dz): The magnitude and direction of the displacement vector are equally important. Larger displacement generally means more work done by a given force component along that direction.
3. Relative Direction (Angle θ): The angle between the force and displacement vectors is crucial. If they are in the same direction (θ=0°), work is maximized (W=|F||D|). If perpendicular (θ=90°), work is zero. If opposite (θ=180°), work is negative and maximized in magnitude (W=-|F||D|). The Work Done by a Force Vector Calculator uses the dot product which inherently includes cos(θ).
4. Frame of Reference: The components of force and displacement depend on the chosen coordinate system. However, the scalar value of work done is independent of the coordinate system if both vectors are described within the same system.
5. Nature of the Force: This calculator assumes a constant force. If the force varies along the path, integration would be required, and this simple dot product formula would only be an approximation or applicable to infinitesimally small displacements.
6. Path of Displacement: For a constant force, the work done depends only on the initial and final positions (the displacement vector), not the path taken. However, for non-constant or non-conservative forces, the path can matter. This calculator assumes work done by a constant force over a straight-line displacement. For more details, see our article on the work-energy theorem.

Understanding these factors helps interpret the results from the Work Done by a Force Vector Calculator.

Frequently Asked Questions (FAQ)

What is work in physics?

In physics, work is the energy transferred to or from an object via the application of force along a displacement. Our Work Done by a Force Vector Calculator quantifies this for constant forces.

What are the units of work?

The standard unit of work (and energy) in the International System of Units (SI) is the Joule (J). 1 Joule = 1 Newton-meter (N·m).

Can work be negative?

Yes, work can be negative. Negative work is done when the force (or its component) acts in the opposite direction to the displacement, meaning the force is trying to slow the object down or remove energy.

What if the force or displacement is zero?

If either the force vector or the displacement vector is zero (all components are zero), the work done is zero. You can see this using the Work Done by a Force Vector Calculator.

What if the force is perpendicular to the displacement?

If the force vector is perpendicular to the displacement vector, the angle θ between them is 90 degrees (or π/2 radians). Since cos(90°) = 0, the work done is zero, regardless of the magnitudes of force and displacement. A force perpendicular to motion does no work (e.g., the magnetic force on a moving charge).

How is this related to the dot product?

The work done by a constant force is defined as the dot product (or scalar product) of the force vector and the displacement vector. Our calculator computes this scalar product work.

Does this calculator work for non-constant forces?

No, this calculator assumes the force is constant over the displacement. For non-constant forces, you would typically need to use integration (W = ∫ F ⋅ ds).

What if my vectors are 2D?

If you have 2D vectors, simply set the Z components (Fz and Dz) to zero in the Work Done by a Force Vector Calculator.

Related Tools and Internal Resources

• Dot Product Calculator: Calculate the dot product of two vectors, which is the basis for work calculation.
• Work-Energy Theorem Explained: Understand the relationship between work done and the change in kinetic energy.
• Scalar Product (Dot Product) of Vectors: Learn more about the mathematical operation used to calculate work.
• Understanding Force: A primer on the concept of force in physics.
• Displacement and Distance: Learn about displacement vectors.
• Kinetic Energy Calculator: Calculate the kinetic energy of an object.