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Basic Concepts and Operations of Vectors

Definition of a Vector

Vector

A vector is a quantity that has both magnitude and direction, usually drawn as an arrow.

Geometric Representation

A vector is represented by a directed segment, written $\vec{AB}$ for a segment from $A$ to $B$ .
The length of a vector is its magnitude (norm), written $|\vec{AB}|$ .
Zero vector: the vector of length 0; its direction is arbitrary.
Unit vector: a vector of length 1.

Coordinate Representation

In 3-space a vector is written

\vec{a} = (a_1, a_2, a_3)

where $a_1, a_2, a_3$ are the projections of the vector onto the $x, y, z$ axes. Concretely, if the vector starts at the origin, its coordinates equal the coordinates of its endpoint.

Vector Addition

Let $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$ . Then

\vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3)

几何解释

Addition can be visualized by the parallelogram law or the triangle rule.

Geometric Meaning

Parallelogram law: place $\vec{a}$ and $\vec{b}$ at a common starting point and build the parallelogram; the diagonal is $\vec{a} + \vec{b}$ .
Triangle rule: move the start of $\vec{b}$ to the end of $\vec{a}$ ; the vector from the start of $\vec{a}$ to the end of $\vec{b}$ is $\vec{a} + \vec{b}$ .

Properties of Addition

Properties of vector addition

Commutativity: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$
Associativity: $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$
Zero vector: $\vec{a} + \vec{0} = \vec{a}$
Opposite: $\vec{a} + (-\vec{a}) = \vec{0}$

Subtraction

Subtraction is addition with the opposite vector:

\vec{a} - \vec{b} = (a_1 - b_1, a_2 - b_2, a_3 - b_3)

Geometrically, if $\vec{a}$ and $\vec{b}$ share a start point, then $\vec{a} - \vec{b}$ is the vector from the endpoint of $\vec{b}$ to the endpoint of $\vec{a}$ .

Scalar Multiplication

For a real number $\lambda$ and $\vec{a} = (a_1, a_2, a_3)$ ,

\lambda\vec{a} = (\lambda a_1, \lambda a_2, \lambda a_3)

几何解释

Multiplying by $\lambda > 0$ stretches or shrinks the vector without changing direction; $\lambda < 0$ reverses the direction.

Properties:

$(\lambda\mu)\vec{a} = \lambda(\mu\vec{a})$
$(\lambda + \mu)\vec{a} = \lambda\vec{a} + \mu\vec{a}$
$\lambda(\vec{a} + \vec{b}) = \lambda\vec{a} + \lambda\vec{b}$
$1 \cdot \vec{a} = \vec{a}$

Linear Combinations

Linear Combination

For vectors $\vec{a}_1, \vec{a}_2, \ldots, \vec{a}_n$ and scalars $k_1, k_2, \ldots, k_n$ , the vector

k_1\vec{a}_1 + k_2\vec{a}_2 + \cdots + k_n\vec{a}_n

is called a linear combination of $\vec{a}_1, \ldots, \vec{a}_n$ .

A useful connection: a vector $\vec{b}$ is collinear with $\vec{a}$ exactly when $\vec{b} = \lambda\vec{a}$ for some scalar $\lambda$ (for nonzero $\vec{a}$ ). Three vectors are coplanar iff one of them is a linear combination of the other two.

Why it matters: every vector in 3-space can be written as a linear combination of the three standard unit vectors $\vec{i}, \vec{j}, \vec{k}$ . This coordinate viewpoint turns geometric problems in space into algebraic ones—the core idea of analytic geometry.

Summary

Symbols Used in This Article

Symbol	Type	Reading/Explanation	Meaning in This Article
$\vec{AB}$	vector	Vector AB	The vector from point $A$ to point $B$
$\vec{a}$	vector	Vector a	A generic vector
$\lvert\vec{a}\rvert$	scalar	norm of a	The magnitude of $\vec{a}$
$\vec{0}$	vector	zero vector	The zero vector
$\lambda$	Greek letter	Lambda	A scalar multiplier

English–Chinese Glossary

English term	Chinese term	Phonetic	Explanation
vector	向量	/ˈvɛktər/	A quantity with magnitude and direction
zero vector	零向量	/ˈzɪərəʊ ˈvɛktər/	The vector of length 0
unit vector	单位向量	/ˈjuːnɪt ˈvɛktər/	A vector of length 1
magnitude/norm	模	/ˈmæɡnɪtjuːd/ /nɔːm/	The length of a vector
vector addition	向量加法	/ˈvɛktər əˈdɪʃən/	The sum of two vectors
scalar multiplication	数乘	/ˈskeɪlə ˌmʌltɪplɪˈkeɪʃən/	Multiplying a vector by a number
linear combination	线性组合	/ˈlɪniə ˌkɒmbɪˈneɪʃən/	A sum of scaled vectors
collinear	共线	/kɒˈlɪniə/	Lying on the same line
coplanar	共面	/kəʊˈpleɪnə/	Lying in the same plane

The Historical Background of Vectors

Discover how vectors were invented through a simple scenario about describing motion.

Products of Vectors

Learn the dot product, cross product, and scalar triple product of vectors, together with their geometric meaning and applications.