3D Force Vector Builder: Angles, Two Points and Sums

3D Force Vector Builder describes a force in space in every way a statics problem gives it: by components, by its three direction angles, by two angles θ and φ, or by two points on its line. Turn the 3D drawing with the mouse or a finger, and add several forces into one resultant.

The numbered parts in Figure 1 are:

1. Picture tabs: each tab is one task; the picture on it shows what the task draws.
2. Drawing: the forces, drawn to scale and updated as you type.
3. Input: the values of the task. Units can be N, kN or lb.
4. Result: the answers, rounded to three significant figures (four when the number starts with 1).
5. How it is worked out: every step of the calculation with your numbers.
6. Scenarios: ready-made real situations; one click fills in the values.

1. Components

Type F_x, F_y and F_z; the magnitude, the unit vector and the direction angles follow.

1. Type the three components.
2. Turn the drawing; the dashed box shows how the components build the force.
3. Switch the angle arcs on or off.

2. Direction angles

Give the magnitude and the three angles, or leave one angle empty: it follows from cos²θ_x + cos²θ_y + cos²θ_z = 1.

1. Type the magnitude and two or three angles.
2. If one angle is empty, choose whether its cosine is positive or negative; both possible values are shown.
3. If all three are given, the app checks that their cos² add up to 1.

3. θ and φ

θ is measured in the xy plane from +x to the projection of the force, φ from +z down to the force.

1. Type the magnitude, θ and φ.
2. The orange dashed line is the projection of the force on the xy plane.
3. The steps split F into F sin φ in the plane and F cos φ along z.

4. Two points

A cable pulls from point A toward point B. The force follows the position vector from A to B.

1. Type the coordinates of A (where the force acts) and B (where it points), and the tension.
2. Read the components of the force and the length of the cable.
3. The steps show head minus tail, the unit vector and F = Fu.

5. Sum of forces

Up to six forces meet at one point; each can be given in any of the four ways.

1. Add forces and choose how each one is given.
2. The table in the steps lists every component and the sums.
3. Read F_R and its direction angles.

6. Scenarios and phones

The scenarios load a guy wire on a mast, the thrust of a drone and the cable of a shop awning. Each scenario opens the right tab and fills in the values, so you can change them and see what happens.

On a phone the drawing comes first and the input, result and steps follow below it. Swipe the picture tabs sideways to see all of them.

7. Tips

• The button with the circular arrow resets the open tab to its starting values; the moon or sun button switches between the light and dark themes.
• Your last values stay in the browser, so the app opens where you left it.
• Angles are measured counterclockwise from +x unless a tab says otherwise.