2D and 3D Vector Calculator

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resultant force calculator

Resultant force and equilibrant force

Open the Resultant of forces mode, enter the magnitude and angle (from the x-axis, counterclockwise) of each force and pick the unit. The table shows Fx and Fy for each one and their sum, and the drawing chains the forces head to tail.

Resolve, add the components and recombine

The universal method for the resultant is to resolve each force into components: Fx = F·cos θ and Fy = F·sin θ, with θ measured from the positive x-axis, counterclockwise. You add the components of all forces, ΣFx and ΣFy, and the resultant vector is R = (ΣFx, ΣFy). Its magnitude is |R| = √(ΣFx² + ΣFy²) and its direction is θ = atan2(ΣFy, ΣFx), placed between 0° and 360°, which already picks the right quadrant. For 50 N at 0°, 30 N at 90° and 20 N at 180°: ΣFx = 50 + 0 - 20 = 30 N and ΣFy = 0 + 30 + 0 = 30 N, so |R| = 30√2 ≈ 42.43 N at 45°.
The equilibrant force is the force that, added to the others, leaves a zero resultant. So E = -R: same magnitude and opposite direction, that is, direction θ + 180°. In the example, E = (-30, -30) N, with 42.43 N at 225°. A body is in translational equilibrium when ΣF = 0, and in that case the calculator says it is in equilibrium and defines no direction for the resultant. Three equal forces of 10 N at 0°, 120° and 240° cancel: the force polygon closes into an equilateral triangle. Two forces of 8 N at 0° and 180° also cancel.
When the angles are multiples of 90° the components are exact and the magnitude comes out with a radical. At other angles the components are decimals rounded to 4 places and the calculator warns that the result is approximate. Two classic examples: 5 N at 0° and 12 N at 90° give 13 N at 67.3801° (the 5-12-13 triangle), and 100 N at 30° with 60 N at 120° give 116.619 N at 60.9638°, because the two are 90° apart and Pythagoras applies. The unit is only a label (N, kN, kgf, lbf or your own): the math works for any vector quantity in the plane, such as velocities, displacements or fields.

Checked examples

Three forces along the axes

Input: 50 N a 0°, 30 N a 90°, 20 N a 180°
Expected output: R = (30, 30) N; |R| = 30√2 ≈ 42,4264 N a 45°

Equilibrant: 225°.

The 5-12-13 triangle

Input: 5 N a 0° e 12 N a 90°
Expected output: R = (5, 12) N; |R| = 13 N a 67,3801°

Equilibrant at 247.3801°.

Equal perpendicular forces

Input: 10 N a 0° e 10 N a 90°
Expected output: |R| = 10√2 ≈ 14,1421 N a 45°

Square diagonal.

Opposite forces

Input: 8 N a 0° e 8 N a 180°
Expected output: R = (0, 0) N; em equilíbrio

No defined direction.

Angles off the axes

Input: 100 N a 30° e 60 N a 120°
Expected output: R ≈ (56,6025; 101,9615) N; |R| ≈ 116,619 N a 60,9638°

Approximate result.

Three forces 120° apart

Input: 10 N a 0°, 120° e 240°
Expected output: R = (0, 0) N; em equilíbrio

The polygon closes.

Full tool FAQ

How do you add and subtract vectors?

Add or subtract component by component: A + B = (a₁ + b₁, a₂ + b₂) and, in 3D, also a₃ + b₃. For A = (3, 4) and B = (-1, 2): A + B = (2, 6) and A - B = (4, 2). Geometrically, the sum is the diagonal of the parallelogram built on A and B, or the vector that closes the chain when you place the tail of B on the tip of A.

How do you calculate the magnitude and the unit vector?

How do you calculate the dot product and the angle between two vectors?

How do you calculate the cross product and the parallelogram area?

What is the triple product and what is it for?

How do you project one vector onto another?

How do you convert magnitude and angle into components?

How do you find the resultant of several forces and the equilibrant force?

How can you tell whether two vectors are parallel, orthogonal or coplanar?

Is the result exact? What are the limits?

Frequently asked questions

How do I measure the angle of each force?

From the positive x-axis, counterclockwise: 0° points right, 90° up, 180° left and 270° down. A 20 N force pointing left is 20 at 180°, never a negative magnitude.

What is the difference between resultant and equilibrant?

The resultant is the single force that replaces the set. The equilibrant is the one that cancels the resultant, with the same magnitude and the opposite direction. If the resultant is already zero there is nothing to balance.

Does the calculator consider the point of application or torque?

No. It treats concurrent forces, applied at the same point, and only adds vectors. Forces at different points also produce torque, which needs another calculation (force times lever arm).

Next steps

Resolve, add the components and recombine

The universal method for the resultant is to resolve each force into components: Fx = F·cos θ and Fy = F·sin θ, with θ measured from the positive x-axis, counterclockwise. You add the components of all forces, ΣFx and ΣFy, and the resultant vector is R = (ΣFx, ΣFy). Its magnitude is |R| = √(ΣFx² + ΣFy²) and its direction is θ = atan2(ΣFy, ΣFx), placed between 0° and 360°, which already picks the right quadrant. For 50 N at 0°, 30 N at 90° and 20 N at 180°: ΣFx = 50 + 0 - 20 = 30 N and ΣFy = 0 + 30 + 0 = 30 N, so |R| = 30√2 ≈ 42.43 N at 45°.

The equilibrant force is the force that, added to the others, leaves a zero resultant. So E = -R: same magnitude and opposite direction, that is, direction θ + 180°. In the example, E = (-30, -30) N, with 42.43 N at 225°. A body is in translational equilibrium when ΣF = 0, and in that case the calculator says it is in equilibrium and defines no direction for the resultant. Three equal forces of 10 N at 0°, 120° and 240° cancel: the force polygon closes into an equilateral triangle. Two forces of 8 N at 0° and 180° also cancel.

When the angles are multiples of 90° the components are exact and the magnitude comes out with a radical. At other angles the components are decimals rounded to 4 places and the calculator warns that the result is approximate. Two classic examples: 5 N at 0° and 12 N at 90° give 13 N at 67.3801° (the 5-12-13 triangle), and 100 N at 30° with 60 N at 120° give 116.619 N at 60.9638°, because the two are 90° apart and Pythagoras applies. The unit is only a label (N, kN, kgf, lbf or your own): the math works for any vector quantity in the plane, such as velocities, displacements or fields.

Checked examples

Three forces along the axes

Input: 50 N a 0°, 30 N a 90°, 20 N a 180°
Expected output: R = (30, 30) N; |R| = 30√2 ≈ 42,4264 N a 45°

Equilibrant: 225°.

The 5-12-13 triangle

Input: 5 N a 0° e 12 N a 90°
Expected output: R = (5, 12) N; |R| = 13 N a 67,3801°

Equilibrant at 247.3801°.

Equal perpendicular forces

Input: 10 N a 0° e 10 N a 90°
Expected output: |R| = 10√2 ≈ 14,1421 N a 45°

Square diagonal.

Opposite forces

Input: 8 N a 0° e 8 N a 180°
Expected output: R = (0, 0) N; em equilíbrio

No defined direction.

Angles off the axes

Input: 100 N a 30° e 60 N a 120°
Expected output: R ≈ (56,6025; 101,9615) N; |R| ≈ 116,619 N a 60,9638°

Approximate result.

Three forces 120° apart

Input: 10 N a 0°, 120° e 240°
Expected output: R = (0, 0) N; em equilíbrio

The polygon closes.

Full tool FAQ

How do you add and subtract vectors?

How do you calculate the magnitude and the unit vector?

How do you calculate the dot product and the angle between two vectors?

How do you calculate the cross product and the parallelogram area?

What is the triple product and what is it for?

How do you project one vector onto another?

How do you convert magnitude and angle into components?

How do you find the resultant of several forces and the equilibrant force?

How can you tell whether two vectors are parallel, orthogonal or coplanar?

Is the result exact? What are the limits?

Frequently asked questions

How do I measure the angle of each force?

From the positive x-axis, counterclockwise: 0° points right, 90° up, 180° left and 270° down. A 20 N force pointing left is 20 at 180°, never a negative magnitude.

What is the difference between resultant and equilibrant?

Does the calculator consider the point of application or torque?

No. It treats concurrent forces, applied at the same point, and only adds vectors. Forces at different points also produce torque, which needs another calculation (force times lever arm).

Continue in the tool

More pages for this tool

Resultant force and equilibrant force

Resolve, add the components and recombine

Checked examples

Three forces along the axes

The 5-12-13 triangle

Equal perpendicular forces

Opposite forces

Angles off the axes

Three forces 120° apart

Full tool FAQ

Frequently asked questions

How do I measure the angle of each force?

What is the difference between resultant and equilibrant?

Does the calculator consider the point of application or torque?

Next steps

Sum, magnitude and direction

Products and angle

Projection

Continue in the tool

More pages for this tool

Resultant force and equilibrant force

Resolve, add the components and recombine

Checked examples

Three forces along the axes

The 5-12-13 triangle

Equal perpendicular forces

Opposite forces

Angles off the axes

Three forces 120° apart

Full tool FAQ

Frequently asked questions

How do I measure the angle of each force?

What is the difference between resultant and equilibrant?

Does the calculator consider the point of application or torque?

Next steps

Sum, magnitude and direction

Products and angle

Projection