Graphical Vector Addition

A vector is a quantity that has both magnitude and direction — velocity, displacement, and force are classic examples. Unlike scalars, you cannot simply add vectors by summing their numbers; the direction matters enormously. Graphical vector addition is a visual method that lets you combine two or more vectors by drawing them as arrows on paper or a screen, where the length of each arrow represents the magnitude and the arrowhead shows the direction. The two fundamental graphical techniques are the triangle law and the parallelogram law. The triangle law states that if two vectors are drawn head-to-tail (the tail of the second placed at the head of the first), the resultant vector is the arrow drawn from the free tail to the free head — closing the triangle. The parallelogram law instead places both vectors tail-to-tail and constructs a parallelogram from them; the resultant is the diagonal drawn from the shared tail. Both methods always give the same resultant, which is a reassuring consistency check. These methods generalise: for three or more vectors, you simply continue placing each new vector head-to-tail to form a polygon. Graphical methods build deep physical intuition about why directions matter. A boat crossing a river and a current pushing it sideways arrive somewhere completely different from where a naive numerical sum would suggest. Before calculators and component methods were widespread, navigators, engineers, and physicists relied heavily on these geometric constructions to solve real problems — and the geometry remains the clearest way to see why vectors behave as they do.

Key takeaways

Representing Vectors as Arrows
Triangle Law — Head-to-Tail Addition
Parallelogram Law — Tail-to-Tail Addition
Polygon Law — Adding Three or More Vectors
Reading Off the Resultant

Representing Vectors as Arrows

Draw each vector as a straight arrow. The length of the arrow must be proportional to the vector's magnitude — choose a scale (e.g. 1 cm = 5 m/s) and stick to it throughout the diagram. The direction of the arrow indicates the vector's direction relative to a reference (usually north or the positive x-axis). Label each arrow with its symbol (e.g. A, B) and annotate the scale.

Triangle Law — Head-to-Tail Addition

To add vector A to vector B using the triangle law: (1) Draw A to scale. (2) Place the tail of B exactly at the head (tip) of A. (3) Draw the resultant R = A + B as the arrow from the tail of A to the head of B. Measure R's length and convert back using your scale to find its magnitude. Use a protractor to measure the angle R makes with a reference direction. The order does not matter: A + B = B + A (vector addition is commutative).

Parallelogram Law — Tail-to-Tail Addition

Place the tails of A and B at the same point. Complete the parallelogram by drawing lines parallel to each vector from the head of the other. The diagonal drawn from the common tail is the resultant R. This method makes symmetry obvious — both A + B and B + A produce the same diagonal. The parallelogram law is especially common in force problems, where two forces act on a single point and you want their combined effect.

Polygon Law — Adding Three or More Vectors

Extend the triangle law: place vector C's tail at the head of B (which was already at the head of A). Continue for as many vectors as you have. The resultant is always the single arrow from the very first tail to the very last head. If the vectors form a closed polygon (head of the last meeting the tail of the first), the resultant is zero — this is the graphical condition for equilibrium.

Reading Off the Resultant

Once you have drawn the resultant arrow: (1) Measure its length with a ruler and divide by your scale factor to get the magnitude. (2) Measure the angle it makes with your chosen reference direction (usually the positive x-axis or north) using a protractor. State your answer as a magnitude with a direction, e.g. '13.4 m/s at 26.6° north of east'. Graphical accuracy depends on careful drawing — small errors in angle or length accumulate, so the method is approximate unless backed up by calculation.

The Key Equation

\vec{R} = \vec{A} + \vec{B}, \quad |\vec{R}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}|\cos\theta}

\vec{R} — Resultant vector (same as component vectors (e.g. N, m/s)): The single vector equivalent in effect to all vectors being added together.
|\vec{A}|, |\vec{B}| — Magnitudes of vectors A and B (same as resultant): The lengths (sizes) of the two vectors being added.
\theta — Angle between the vectors (degrees or radians): The angle between the two vectors when placed tail-to-tail. When θ = 90°, cos θ = 0 and the Pythagorean theorem is recovered; when θ = 0° the vectors are parallel and R = A + B.

Worked Example

Problem. A hiker walks 40 m due east, then 30 m due north. Using the triangle law, find the magnitude and direction of the resultant displacement.

|\vec{R}| = \sqrt{40^2 + 30^2} = 50 \text{ m}, \quad \phi = \arctan\!\left(\frac{30}{40}\right) \approx 36.9°

Answer. The resultant displacement is 50 m at 36.9° north of east.

Figure

Diagram

Origins · 1 minSimon Stevin and how this was discovered

Simon Stevin

1548–1620 · Flemish (Dutch)

Simon Stevin was the first to clearly demonstrate the parallelogram law for the composition of forces, publishing his proof in 'De Beghinselen der Weeghconst' (1586). Using a celebrated thought experiment with a chain of balls draped over a triangular prism, he showed geometrically that forces combine according to the parallelogram rule — a foundational insight for all of mechanics and the direct ancestor of modern graphical vector addition. Stevin also championed the use of decimal fractions in everyday arithmetic and designed a sail-powered land yacht that reportedly carried 28 passengers along the Dutch beach faster than a galloping horse — making him one of history's more adventurous physicist-engineers.

Vector and Scalar Quantities — Graphical addition only makes sense once you understand what distinguishes a vector (magnitude + direction) from a scalar — this concept provides that essential foundation.
Component Resolution — Resolving vectors into perpendicular components is the algebraic counterpart to graphical addition; together they form the complete toolkit for vector arithmetic.
Displacement and Velocity Vectors — Displacement and velocity are the most common vectors students add graphically, making them the natural application context for the triangle and parallelogram laws.
Resultant Force and Equilibrium — The parallelogram law is used directly to find resultant forces; equilibrium occurs when the vector polygon closes — a concept built entirely on graphical addition.