1 Introduction to vectors
Vectors are fundamental objects in linear algebra, defined by both magnitude (length) and direction. They are used extensively in mathematics and applied sciences to represent quantities that possess these two attributes. The arithmetic of vectors—addition, subtraction, and scalar multiplication—provides a set of rules for combining and manipulating these objects.
1.1 Definition and notation
A vector is typically denoted by a boldface letter (e.g., v) or with an arrow over the letter (e.g., \(\vec{v}\)). In coordinate geometry, a vector in \(n\)-dimensional space is represented as an ordered \(n\)-tuple of real numbers, such as \(\mathbf{v} = (v_1, v_2, \ldots, v_n)\). The components \(v_i\) correspond to the vector's displacement along each axis. The set of all vectors with \(n\) components, together with the operations defined in this article, forms a vector space.
1.2 Geometric interpretation
Geometrically, a vector is visualized as an arrow with a specific length and direction. The tail of the arrow represents the starting point, and the head points to the destination. Two vectors are considered equal if they have the same magnitude and direction, regardless of their position in space. This interpretation is especially useful in physics and engineering, where vectors represent forces, velocities, or displacements.
1.3 Coordinate representation
| In a Cartesian coordinate system, a vector is expressed in terms of its components along orthogonal axes. For example, in two dimensions, \(\mathbf{v} = (x, y)\); in three dimensions, \(\mathbf{v} = (x, y, z)\). The magnitude (or norm) is given by \(\|\mathbf{v}\| = \sqrt{x^2 + y^2}\) (in 2D) or \(\sqrt{x^2 + y^2 + z^2}\) (in 3D). The direction is described by the angles the vector makes with the coordinate axes. |
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2 Basic operations
The core operations of vector arithmetic are addition, subtraction, and scalar multiplication. These operations satisfy a set of algebraic properties that make vectors a versatile mathematical tool.
2.1 Vector addition
Vector addition combines two vectors to produce a third vector. There are several equivalent ways to define and visualize this operation.
2.1.1 Parallelogram rule
Place the tails of the two vectors at the same point. Construct a parallelogram with the vectors as adjacent sides. The diagonal from the common tail to the opposite vertex represents the sum of the two vectors. This rule emphasizes the geometric nature of addition.
2.1.2 Triangle rule
Place the tail of the second vector at the head of the first. The vector from the tail of the first to the head of the second is the sum. This method is particularly intuitive for representing consecutive displacements: walking along one vector and then the next yields a net displacement given by the sum.
2.1.3 Component-wise addition
In coordinate form, vector addition is performed by adding corresponding components. For \(\mathbf{u} = (u_1, u_2, \ldots, u_n)\) and \(\mathbf{v} = (v_1, v_2, \ldots, v_n)\), their sum is \(\mathbf{u} + \mathbf{v} = (u_1+v_1, u_2+v_2, \ldots, u_n+v_n)\). This aligns with the parallelogram and triangle rules and provides a straightforward algebraic method.
2.2 Vector subtraction
Vector subtraction finds the difference between two vectors, effectively the displacement from one vector's endpoint to the other's.
2.2.1 Definition as addition of the negative
The negative of a vector \(\mathbf{v}\), denoted \(-\mathbf{v}\), has the same magnitude but opposite direction. Subtraction is then defined as \(\mathbf{u} - \mathbf{v} = \mathbf{u} + (-\mathbf{v})\). Geometrically, this corresponds to adding the reversed version of the second vector.
2.2.2 Component-wise subtraction
Similar to addition, subtraction is performed component‑wise: \(\mathbf{u} - \mathbf{v} = (u_1 - v_1, u_2 - v_2, \ldots, u_n - v_n)\). This operation yields a vector that points from the head of \(\mathbf{v}\) to the head of \(\mathbf{u}\) when both tails are placed at the same origin.
2.3 Scalar multiplication
Scalar multiplication multiplies a vector by a real number (a scalar), altering its length while preserving (or reversing) its direction.
2.3.1 Scaling a vector
| For a scalar \(c\) and vector \(\mathbf{v} = (v_1, v_2, \ldots, v_n)\), the product is \(c\mathbf{v} = (c v_1, c v_2, \ldots, c v_n)\). If \( | c | > 1\), the vector lengthens; if \(0 < | c | < 1\), it shortens. The direction remains unchanged when \(c > 0\). |
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2.3.2 Negative scalar and direction reversal
When \(c\) is negative, the vector's direction is reversed. For example, \(-2\mathbf{v}\) points opposite to \(\mathbf{v}\) and is twice as long. This operation is essential for representing opposite forces or reverse displacements.
3 Properties of vector arithmetic
The operations defined above obey a set of algebraic laws analogous to those for real numbers, making vectors a well‑behaved algebraic structure.
3.1 Commutativity and associativity of addition
Vector addition is commutative: \(\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\). It is also associative: \((\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})\). These properties follow directly from component‑wise addition and the commutativity and associativity of real‑number addition.
3.2 Distributive laws
Distributive laws link scalar multiplication with addition, enabling the expansion of expressions involving scalars and vectors.
3.2.1 Scalar distribution over vector addition
For a scalar \(c\) and vectors \(\mathbf{u}\) and \(\mathbf{v}\), \(c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}\). This holds because each component is multiplied by \(c\) before adding.
3.2.2 Scalar distribution over scalar addition
For scalars \(a\) and \(b\) and a vector \(\mathbf{v}\), \((a + b)\mathbf{v} = a\mathbf{v} + b\mathbf{v}\). This reflects the distributive property of multiplication over addition in the real numbers.
3.3 Existence of additive identity and inverses
The zero vector \(\mathbf{0} = (0,0,\ldots,0)\) serves as the additive identity: \(\mathbf{v} + \mathbf{0} = \mathbf{v}\) for every vector \(\mathbf{v}\). Every vector \(\mathbf{v}\) has an additive inverse \(-\mathbf{v}\) such that \(\mathbf{v} + (-\mathbf{v}) = \mathbf{0}\). These properties are crucial for solving vector equations.
4 Vector arithmetic in different contexts
The same operations apply to vectors of any dimension, though geometric intuition dims as dimension increases.
4.1 Two-dimensional vectors
In the plane, vectors are often written as \((x, y)\) and visualized as arrows. The arithmetic is directly linked to plane geometry and trigonometry. Applications include planar motion, 2D computer graphics, and complex numbers (where a complex number \(a+bi\) can be treated as a 2D vector).
4.2 Three-dimensional vectors
Three‑dimensional vectors \((x, y, z)\) are standard in physics for describing spatial quantities. The operations are identical in form but involve three components. The cross product, a related operation, exists only in 3D (and 7D) and is often introduced alongside the arithmetic covered here.
4.3 Higher-dimensional vectors
Vectors with four or more components are abstract, but their arithmetic remains the same: component‑wise operations. Such vectors appear in data science (e.g., feature vectors), relativity (spacetime four‑vectors), and signal processing (discrete signals as high‑dimensional vectors). The algebraic properties continue to hold, ensuring consistency across dimensions.
5 Applications
Vector arithmetic is a practical tool in numerous fields, providing a language for describing quantities with both magnitude and direction.
5.1 Physics: displacement and velocity
Displacement is a vector from an initial to a final position; velocity is the rate of change of displacement. Adding velocity vectors yields resultant velocities (e.g., an airplane's velocity relative to the ground from its airspeed and wind velocity). Scalar multiplication represents changes in speed or direction reversal.
5.2 Computer graphics: transformations
In 2D and 3D graphics, vectors represent positions, directions, and colors. Vector addition shifts objects (translation), and scalar multiplication scales them. Combined with matrices (which represent rotations and scaling), vector arithmetic forms the basis of geometric transformations used in rendering and animation.
5.3 Engineering: force analysis
Forces are vectors. Engineers use vector addition (often via the parallelogram rule) to find net forces on structures or objects. Scalar multiplication scales forces when applying multiple similar loads. Subtraction determines unbalanced forces, guiding design decisions in bridges, machines, and vehicles.