Force components

How to split a force into a horizontal and a vertical part, and put it back together

Last updated 2026-09-25

Any force in a plane can be replaced by two forces at right angles, one horizontal (Fx) and one vertical (Fy), that together do exactly the same job. For a force F at angle θ from the horizontal, Fx = F·cos θ and Fy = F·sin θ. To go back, F = √(Fx² + Fy²).

One arrow, two numbers

A force has a size and a direction. The arrow shows both at once, but arrows are hard to add up. Numbers are easy.

So we ask two simpler questions instead:

• How hard does this force push sideways? That's Fx.
• How hard does it push up or down? That's Fy.

Think of the force's shadow. With the sun straight overhead, the arrow's shadow on the ground is its horizontal component. With the light coming from the side, its shadow on a wall is the vertical component. The two shadows together describe the arrow completely.

The components replace the force; they don't add to it. Either you use the arrow, or you use its two components, never both.

From a force to its components

Measure the angle θ from the horizontal:

Fx = F · cos θ

Fy = F · sin θ

Example: a 10 kN force, 30° above the horizontal, pointing right.

Fx = 10 × cos 30° = 8.66 kN

Fy = 10 × sin 30° = 5.00 kN

Notice that 8.66 + 5.00 isn't 10. Components don't add up to the force by simple addition; they add up the way the sides of a right triangle do.

From components back to the force

The force is the hypotenuse of the triangle its components make:

F = √(Fx² + Fy²)

Example: Fx = 5 kN, Fy = −8 kN.

F = √(5² + 8²) = √89 = 9.43 kN

The size of the force always comes out positive. The direction comes from the signs of the components.

Signs tell the direction

Truzme, like most statics textbooks, uses x to the right and y upward.

So −8 kN doesn't mean "negative force". It means 8 kN downward. Gravity loads are almost always a negative Fy.

No angle? Use the slope

In a truss, you rarely know angles. You know where the nodes are. A member that goes 3 m across and 4 m up is 5 m long, so a force along it splits the same way:

Fx = F × 35

Fy = F × 45

Across ÷ length and up ÷ length are just cos θ and sin θ, without ever working out θ. That's exactly how the support reactions example splits its 5 kN members into 3 kN and 4 kN.

Why Truzme shows components

In Truzme, each load and reaction is drawn as one arrow, and labelled with its two components instead of its size. A load labelled 5 kN and −8 kN pushes 5 kN to the right and 8 kN down.

It's labelled this way because that's how you use the numbers. Balance is checked one direction at a time:

ΣFx = 0

ΣFy = 0

To add forces, you add their components, x with x and y with y:

(5, −8) + (−2, 3) = (3, −5) kN

Adding the sizes (9.43 + 3.61) would give you a number that means nothing. If you do need the size of a force, work it out from the two components, as above. But the two numbers on the label are the ones that go into your equations.

Common mistakes

• Adding the sizes of forces. Only forces in exactly the same direction add like that. Everything else goes through components.
• Swapping sin and cos. It depends on where you measure the angle from. Measured from the horizontal, cos gives Fx. Measured from the vertical, cos gives Fy. Draw the triangle and check: the component next to the angle gets cos.
• Calculator in radians. cos 30 in radians is 0.154, not 0.866. If a component comes out tiny or negative for no reason, check the mode.
• Dropping the sign. A downward component is negative. Losing a minus sign is the most common reason a hand calculation doesn't match Truzme.
• Using the components and the force together. On a free-body diagram, draw one or the other. Drawing both counts the force twice.

Try it in Truzme

Add a nodal load in Truzme and type its x and y values; the arrow follows. Set only y for a gravity load, or both for a load at an angle. The Getting started guide walks you through it.