Dot Product Visualizer: Angles and Projections in 3D

Dot Product Visualizer shows what the dot product means in statics: the angle between two directions and the part of a force along a line. Each result is drawn in 3D, with the parallel part, the perpendicular part and a right-angle mark.

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. Angle between

A · B = AB cos θ gives the angle between two vectors, or between two lines that meet at a point.

1. Choose Two vectors and type their components, or Three points and type O, A and B.
2. Read the dot product and the angle θ.
3. A negative dot product means an angle larger than 90°.

2. Projection

The part of a force along a line is F · u, where u is the unit vector of the line.

1. Give the force by its components or by its magnitude and two points.
2. Give the line by a direction or by two points.
3. Read F_∥, the angle with the line and F_⊥.

3. Parallel and perpendicular

The force is split into two vectors: F_∥ along the line and F_⊥ across it.

1. Enter the force and the line as in the projection tab.
2. Read both vectors in Cartesian form.
3. The last step checks that they are perpendicular and that their squares add up to F².

4. Angle with a plane

The angle between a line and a plane is 90° minus the angle between the line and the plane normal.

1. Give the line by a direction or by two points.
2. Give the plane by its normal or by three points on it.
3. Read β, the angle with the plane, and α, the angle with the normal.

5. Scenarios and phones

The scenarios load a cable pulling on a pipe, a rope pulling a cart along a rail and wind pressing on a tilted solar panel. 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.

6. 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.