What it does
A vector has a size and a direction: a force, a velocity, a walk across a field. Two vectors do not add like ordinary numbers; 3 steps east and 4 steps north take you only 5 steps from where you started. This tool lets you drag two arrows, A (blue) and B (orange), and draws their sum, the resultant R (red), in three ways:
- Triangle: B starts where A ends; R joins the start of A to the end of B.
- Parallelogram: A and B start at the same point; R is the diagonal of the parallelogram they make.
- Components: each arrow is split into a part along x and a part along y; the parts simply add.
All three give the same arrow. Below the picture you get the size and direction of R from the law of cosines and from components, with your numbers filled in. A second card hangs a weight from two strings and uses Lami’s theorem to find the tension in each string, which is three forces in balance.
It follows Class 11 Physics, motion in a plane (scalars and vectors, adding and subtracting vectors, resolving a vector into components) and the equilibrium of concurrent forces from the laws of motion. Before this, equations of motion covers motion along a line; after it, the same adding splits the forces on a block in friction ramp, and 3D vector lab takes vectors into three dimensions.
How to use it
- Drag the round handle at the tip of A or of B, or use the size and direction sliders. Directions are measured anticlockwise from the right (east), as on a maths graph.
- Pick Triangle, Parallelogram or Components to change how the sum is drawn.
- Tick Also show A − B to see the difference as a dashed arrow. Subtracting B is the same as adding B turned round.
- In the lower card, set the weight and the angle of each string to the ceiling. The arrows show the two tensions and the weight; their sizes are worked out with Lami’s theorem and checked by balancing the forces across and up.
Worked example
Open the page: A = 6 at 0° and B = 4 at 60°, so the angle between them is θ = 60°. The law of cosines gives R = √(36 + 16 + 2 × 6 × 4 × 0.5) = √76 = 8.72. By components, A = (6, 0) and B = (2, 3.46), so R = (8, 3.46), which also has size 8.72 and points 23.4° above A.
In the Lami card a 100 N weight hangs from strings at 30° and 60° to the ceiling. The strings are then 90° apart, and Lami’s theorem gives T₁ = 100 × sin 150° ÷ sin 90° = 50 N and T₂ = 100 × sin 120° ÷ sin 90° = 86.6 N. Check: across, 50 cos 30° = 43.3 N and 86.6 cos 60° = 43.3 N cancel; up, 25 + 75 = 100 N holds the weight.
Tips
- The resultant of A and B is at most A + B (same direction) and at least |A − B| (opposite directions).
- When A and B are at 90°, the law of cosines becomes Pythagoras’ theorem, because cos 90° = 0.
- In Lami’s theorem each force goes with the sine of the angle between the other two forces. A steeper string carries more of the load.
FAQ
Why does the triangle law give the same answer as the parallelogram law? The far side of the parallelogram is just B moved, without turning it, to start at the tip of A. Moving a vector without turning it does not change it.
Can I add three or more vectors? Yes: add the x parts, add the y parts, and the resultant is made from those two totals. That is what the Components view shows for two.
Is anything stored? No. The questions are made in your browser and nothing is saved or sent.
Where the numbers come from
- Vector addition by components, the law of cosines R² = A² + B² + 2AB cos θ, tan α = B sin θ / (A + B cos θ) and Lami's theorem for three concurrent forces in equilibrium: standard school physics found in any textbook.
- Checked against an independent component sum on 1000 random pairs of vectors, and Lami's tensions against a direct horizontal and vertical force balance on 1000 random string angles.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.