Vector Addition and Subtraction Calculator

Math Linear Algebra • Vectors and Basic Operations

Written by STEM Calculators Team Published February 21, 2026 Updated February 24, 2026

Add or subtract vectors component-wise in \(\mathbb{R}^n\) and visualize the result using the parallelogram law and head-to-tail method (2D/3D).

Arrow sum plot

Enter vectors to draw. 2D: drag to pan, wheel to zoom. 3D: drag to rotate, Shift+drag to pan, wheel to zoom.

\(\mathbf{V}\) \(\mathbf{W}\) Result head-to-tail (dashed)

Enter vectors and click “Calculate”.

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Theory

Vector addition and subtraction in \(\mathbb{R}^n\)

A vector in \(\mathbb{R}^n\) is an ordered list of \(n\) real numbers: \[ \mathbf{V} = (v_1, v_2, \dots, v_n). \] You can think of it as a displacement, a force, a velocity, or simply a point from the origin in an \(n\)-dimensional space.

Component-wise (algebraic) definitions

\[ \mathbf{V}+\mathbf{W}=(v_1+w_1,\; v_2+w_2,\; \dots,\; v_n+w_n), \] \[ \mathbf{V}-\mathbf{W}=(v_1-w_1,\; v_2-w_2,\; \dots,\; v_n-w_n). \]

Subtraction is the same as adding the additive inverse: \[ \mathbf{V}-\mathbf{W}=\mathbf{V}+(-\mathbf{W}),\qquad -\mathbf{W}=(-w_1,\dots,-w_n). \]

Geometric interpretation: head-to-tail and parallelogram law

Head-to-tail method (2D/3D intuition)

To add vectors, draw \(\mathbf{V}\) from the origin, then draw \(\mathbf{W}\) starting at the head (tip) of \(\mathbf{V}\). The vector from the origin to the final head is \(\mathbf{V}+\mathbf{W}\).

Parallelogram law

If \(\mathbf{V}\) and \(\mathbf{W}\) are drawn from the same tail (origin), the diagonal of the parallelogram formed by translating \(\mathbf{V}\) parallel to \(\mathbf{W}\) (and vice versa) is the sum \(\mathbf{V}+\mathbf{W}\).

For subtraction, visualize \(\mathbf{V}-\mathbf{W}\) as \(\mathbf{V}+(-\mathbf{W})\): reverse the direction of \(\mathbf{W}\) and then add.

Core properties (why vector addition behaves “nicely”)

Commutative: \(\mathbf{V}+\mathbf{W}=\mathbf{W}+\mathbf{V}\).
Associative: \((\mathbf{U}+\mathbf{V})+\mathbf{W}=\mathbf{U}+(\mathbf{V}+\mathbf{W})\).
Zero vector: there exists \(\mathbf{0}=(0,\dots,0)\) with \(\mathbf{V}+\mathbf{0}=\mathbf{V}\).
Additive inverse: for each \(\mathbf{V}\) there is \(-\mathbf{V}\) with \(\mathbf{V}+(-\mathbf{V})=\mathbf{0}\).

These axioms are the foundation of “vector spaces” in linear algebra: addition is closed, consistent, and reversible.

Applied example: resultant force

If two forces act on an object, \(\mathbf{F}_1\) and \(\mathbf{F}_2\), the net (resultant) force is \[ \mathbf{F}_{\text{net}} = \mathbf{F}_1 + \mathbf{F}_2. \] In 2D, this matches the parallelogram law used in physics diagrams. In 3D, the same component-wise rule applies.

University note: beyond finite dimensions

In more advanced courses, vectors can be functions, sequences, or signals (infinite-dimensional objects). The “addition” rule is still component-wise in spirit: add values point-by-point (or term-by-term). Extra structure (like norms and convergence) is used to ensure everything behaves well.

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