Vector Operations: Addition, Subtraction & Scaling

Vector Addition

Adding two vectors means pairing up their components and summing each pair independently. For

\mathbf{a} = (a_1, a_2, \ldots, a_n)

and

\mathbf{b} = (b_1, b_2, \ldots, b_n)

in the same

\mathbb{R}^n

:

Vector Addition

\mathbf{a} + \mathbf{b} = (a_1 + b_1,\ a_2 + b_2,\ \ldots,\ a_n + b_n)

Learn more about this formula: Vector Addition →

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The result is again a vector in

\mathbb{R}^n

— addition does not change the dimension. Both inputs must belong to the same space; adding a vector in

\mathbb{R}^2

to a vector in

\mathbb{R}^3

is undefined because there is no way to match up the components.

Geometrically, vector addition has two equivalent visualizations. In the tip-to-tail method, the tail of

\mathbf{b}

is placed at the head of

\mathbf{a}

, and the sum

\mathbf{a} + \mathbf{b}

is the vector running from the tail of

\mathbf{a}

to the head of

\mathbf{b}

. In the parallelogram method, both vectors share a common tail, and the sum is the diagonal of the parallelogram they form. The two constructions always yield the same result, and each highlights a different aspect of addition: tip-to-tail emphasizes sequential displacement, while the parallelogram makes the symmetry between

\mathbf{a}

and

\mathbf{b}

visually explicit.

u + v, every component settled

Each slot of the result was filled from the matching pair above it and nothing else — the third entry never consulted the first. That independence is the whole reason addition needs both vectors to have the same length, and the reason it costs one addition per component. Step through it slot by slot, or switch to subtraction, on the vector addition visualizer.

Because the components never interact, everything else about addition — its commutativity, its associativity, the zero vector — follows from the arithmetic of a single slot.

Vector Notation

Notation

Vector Notation

How print and blackboard mark a vector differently, what the components and brackets are doing, and the superscript that names a whole space.

\mathbf{v}

·

\vec{v}

The vector v

The marking exists to separate vectors from scalars at a glance. Print uses bold — this site's convention; handwriting cannot bold, so blackboards and physics texts draw the arrow. Same object, one glyph per medium.

CasesThe zero vector inherits the mark:

\mathbf{0}

in Properties of Addition below is a vector, while plain

0

is a number — one equation can legally hold both, and only the typeface tells them apart.

Also writtenUnderlined

\underline{v}

— the older typescript convention, a typist's instruction to the printer to set bold; still seen in some European texts.

Do not confuseA scalar. Dropping the mark mid-computation is the commonest linear-algebra writing error —

v

and

\mathbf{v}

answer to different algebra, and

\frac{\mathbf{a}}{\mathbf{b}}

is meaningless: vectors do not divide.

\mathbf{v} = (v_1, \ldots, v_n)

v with components v-one through v-n

The subscript names a coordinate slot —

v_2

is the second component of one vector, a scalar. Another job for the subscript, alongside the sequence position and root enumeration uses.

CasesThe tuple

(v_1, \ldots, v_n)

and the column

\begin{bmatrix} v_1 \\ \vdots \\ v_n \end{bmatrix}

carry the same data; the column form becomes obligatory once matrices enter, because matrix multiplication reads shapes. Row versus column is a real distinction there, not typography.

Also written

\langle v_1, v_2 \rangle

— angle-bracket components in some American calculus texts; risky, since the same brackets later mean the inner product.

Do not confuseA point.

(3, 4)

names both the point and the vector from the origin to it — same glyphs, and only context says whether it is a location or a displacement.

\mathbb{R}^n

R n — real n-space

Blackboard-bold

\mathbb{R}

for the real numbers, superscript

n

for how many coordinates:

\mathbb{R}^n

is the space of all

n

-component real vectors. The membership statement

\mathbf{v} \in \mathbb{R}^3

is the standard way to declare a vector's size.

Cases

\mathbb{R}^2

is the plane,

\mathbb{R}^3

ordinary space. Operations require matching spaces — the same-dimension rule in Vector Addition above is written as “both in

\mathbb{R}^n

”.

Also written

\mathbb{C}^n

for complex components; the blackboard-bold letter family itself (

\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}

) is catalogued with the set-theory symbols.

Do not confuseA power of a number. The superscript is a Cartesian power —

n

copies of

\mathbb{R}

glued into tuples — not

\mathbb{R}

multiplied by itself; there is nothing to multiply.

All linear algebra symbols →↑ Vectors

Properties of Addition

Vector addition obeys four algebraic rules that govern how sums behave. Each has a geometric counterpart that can be verified by drawing the vectors involved.

Commutativity

\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}

The order in which two vectors are added does not affect the result. Geometrically, the parallelogram has the same diagonal regardless of which vector is placed first. At the component level, this follows directly from commutativity of real number addition:

a_i + b_i = b_i + a_i

for every component.

Associativity

(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})

When adding three or more vectors, grouping does not matter. The tip-to-tail construction confirms this: chaining

\mathbf{a}

,

\mathbf{b}

, and

\mathbf{c}

end to end produces the same resultant vector no matter which pair is summed first. This property allows sums of multiple vectors to be written without parentheses.

Additive Identity

\mathbf{a} + \mathbf{0} = \mathbf{a}

The zero vector

\mathbf{0} = (0, 0, \ldots, 0)

leaves any vector unchanged under addition. It functions as the neutral element: adding it contributes nothing to any component. Geometrically, the zero vector is a point with no length and no direction — appending it tip-to-tail adds no displacement.

Additive Inverse

\mathbf{a} + (-\mathbf{a}) = \mathbf{0}

Every vector

\mathbf{a}

has a corresponding vector

-\mathbf{a} = (-a_1, -a_2, \ldots, -a_n)

that cancels it exactly. The inverse has the same magnitude as

\mathbf{a}

but points in the opposite direction. Adding a vector to its inverse returns to the origin — the displacements undo each other completely.

Property	Formula	Note
Commutativity	a + b = b + a	order does not affect the sum
Associativity	(a + b) + c = a + (b + c)	grouping does not matter; parentheses become optional
Additive identity	a + 0 = a	the zero vector leaves any vector unchanged
Additive inverse	a + (−a) = 0	every vector has a unique negation, same length, opposite direction

Vector Subtraction

Subtraction is not an independent operation — it is addition combined with negation. The difference

\mathbf{a} - \mathbf{b}

is defined as:

Vector Subtraction

\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b}) = (a_1 - b_1,\ a_2 - b_2,\ \ldots,\ a_n - b_n)

Learn more about this formula: Vector Subtraction →

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Reducing subtraction to addition and negation means that all four properties of addition — commutativity, associativity, identity, and inverse — carry over automatically. No separate set of rules is needed.

The geometric picture of subtraction is particularly useful. When

\mathbf{a}

and

\mathbf{b}

are drawn from a common tail, the difference

\mathbf{a} - \mathbf{b}

is the vector pointing from the tip of

\mathbf{b}

to the tip of

\mathbf{a}

. This interpretation connects subtraction directly to distance: the length of

\mathbf{a} - \mathbf{b}

is the Euclidean distance between the heads of the two vectors, formalized as

d(\mathbf{a}, \mathbf{b}) = \|\mathbf{a} - \mathbf{b}\|

on the magnitude page.

Scalar Multiplication

Scalar multiplication takes a real number

c

and a vector

\mathbf{a}

and scales every component by

c

:

Scalar Multiplication of Vectors

c\mathbf{a} = (ca_1,\ ca_2,\ \ldots,\ ca_n)

Learn more about this formula: Scalar Multiplication of Vectors →

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The result is always a vector in the same

\mathbb{R}^n

. Geometrically, scalar multiplication changes the length of the vector by a factor of

|c|

while preserving or reversing its direction depending on the sign of

c

.

When

c > 0

, the scaled vector

c\mathbf{a}

points in the same direction as

\mathbf{a}

. If

c > 1

, the vector stretches; if

0 < c < 1

, it compresses. When

c < 0

, the direction flips — the scaled vector points opposite to

\mathbf{a}

, with its length multiplied by

|c|

. The boundary case

c = 0

collapses any vector to the zero vector:

0\mathbf{a} = \mathbf{0}

.

Because scalar multiplication only ever stretches, compresses, or reverses a vector along its own line, the result

c\mathbf{a}

is always parallel to

\mathbf{a}

(provided

\mathbf{a} \neq \mathbf{0}

). This observation becomes important later: two vectors are parallel precisely when one is a scalar multiple of the other.

Case	Effect on length	Effect on direction
c > 1	stretches by factor \|c\|	same as a
0 < c < 1	compresses by factor \|c\|	same as a
c < 0	scales by factor \|c\|	reversed (opposite to a)
c = 0	collapses to zero	undefined — result is the zero vector

cv, every component scaled

One number has reached every entry: each component of the result is the matching component multiplied by the same scalar. Nothing has been mixed between slots, which is why scaling changes a vector's length and possibly its direction but never tilts it off its own line. Try a negative scalar and watch the arrow flip on the scalar multiplication visualizer.

Scaling and adding are the only two operations in play here, and the rest of this page is a catalogue of how they behave together.

Properties of Scalar Multiplication

Scalar multiplication satisfies its own set of algebraic rules that describe how scalars and vectors interact. Combined with the properties of addition, these rules form the complete algebraic framework for working with vectors.

Associativity with Scalars

c(d\mathbf{a}) = (cd)\mathbf{a}

Scaling a vector by

d

and then by

c

produces the same result as scaling once by the product

cd

. The operations collapse into a single multiplication on the scalar side.

Distributivity over Vector Addition

c(\mathbf{a} + \mathbf{b}) = c\mathbf{a} + c\mathbf{b}

A scalar applied to a sum distributes across both terms. Geometrically, scaling the diagonal of a parallelogram by

c

yields the same vector as scaling both sides by

c

and then forming the new parallelogram.

Distributivity over Scalar Addition

(c + d)\mathbf{a} = c\mathbf{a} + d\mathbf{a}

Two scalars summed before multiplication produce the same result as two separate scalings added afterward. This rule links the arithmetic of real numbers to the algebra of vectors.

Multiplicative Identity

1\mathbf{a} = \mathbf{a}

Scaling by

1

leaves a vector unchanged. This ensures that the scalar

1

acts neutrally, just as the zero vector acts neutrally under addition.

Consequences

Two special cases follow immediately from these rules. Setting

c = 0

gives

0\mathbf{a} = \mathbf{0}

: scaling any vector by zero produces the zero vector. Setting

c = -1

gives

(-1)\mathbf{a} = -\mathbf{a}

: scaling by

-1

produces the additive inverse. Neither fact requires a separate axiom — both are consequences of the properties above.

Property	Formula	Note
Associativity with scalars	c(da) = (cd)a	repeated scalings collapse to a single multiplication
Distributivity over vector addition	c(a + b) = ca + cb	a scalar spreads across each term of a vector sum
Distributivity over scalar addition	(c + d)a = ca + da	a sum of scalars splits into two separate scalings of the same vector
Multiplicative identity	1a = a	scaling by 1 leaves any vector unchanged

The Algebraic Foundation

Addition and scalar multiplication are not just two operations among many — they are the two operations on which the entire algebraic theory of vectors rests. Every other construction in this section is built from them. The dot product multiplies corresponding components and sums the results — a sequence of scalar multiplications followed by real-number addition. The cross product combines components through differences of pairwise products — again, scalar multiplication and addition in a specific pattern. A linear combination

c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k

is nothing more than repeated scalar multiplication followed by repeated addition.

The ten properties listed on this page — four for addition, four for scalar multiplication, plus two distributive laws — are not arbitrary. They are precisely the axioms that define a vector space. Any collection of objects that satisfies these rules qualifies as a vector space, whether the objects are arrows in

\mathbb{R}^3

, polynomials, matrices, or functions. The vectors in

\mathbb{R}^n

studied throughout this section are the most concrete example, but the algebraic structure they exhibit extends far beyond ordered tuples of numbers.

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Vectors · algebraic foundation

The eight laws the operations obey

Four govern addition, four govern scalar multiplication. Together with closure they are exactly the vector space axioms — which is why anything proved from them holds for polynomials and matrices too, not just arrows.

08laws

Unconditional8

i

Commutativity of addition§ 2

\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}

Order of addition does not matter.

Unconditional

Requires: component-wise, so it inherits from $\mathbb{R}$

Order of addition does not matter. Geometrically, the parallelogram is the same figure traversed two ways — tip-to-tail in either order lands on the same diagonal.

ii

Associativity of addition§ 2

(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})

Grouping does not matter, so a chain of additions needs no parentheses.

Unconditional

Requires: any three vectors in the same $\mathbb{R}^n$

Grouping does not matter, so a chain of additions needs no parentheses. This is what lets linear combinations be written as a single sum.

iii

Additive identity§ 2

\mathbf{a} + \mathbf{0} = \mathbf{a}

The zero vector leaves everything unchanged.

Unconditional

Requires: $\mathbf{0}$ the vector of all zeros

The zero vector leaves everything unchanged. It is the origin — the one point every subspace must contain, since without it there is no identity to add.

iv

Additive inverse§ 2

\mathbf{a} + (-\mathbf{a}) = \mathbf{0}

Every vector can be undone by the arrow pointing the opposite way.

Unconditional

Requires: one inverse per vector

Every vector can be undone by the arrow pointing the opposite way. This is what makes subtraction definable at all — $\mathbf{a} - \mathbf{b}$ is shorthand for $\mathbf{a} + (-\mathbf{b})$ , not a separate operation.

v

Associativity with scalars§ 5

c(d\mathbf{a}) = (cd)\mathbf{a}

Scaling twice equals scaling once by the product.

Unconditional

Requires: any scalars $c, d$

Scaling twice equals scaling once by the product. Note the two multiplications are different operations — $cd$ happens between numbers, $c(d\mathbf{a})$ between a number and a vector — and the law is the claim that they agree.

vi

Distributivity over vector addition§ 5

c(\mathbf{a} + \mathbf{b}) = c\mathbf{a} + c\mathbf{b}

Scaling spreads over a sum of vectors.

Unconditional

Requires: any scalar $c$

Scaling spreads over a sum of vectors. Together with the next law this is precisely what linearity demands of a transformation — the definition is these two conditions and nothing more.

vii

Distributivity over scalar addition§ 5

(c + d)\mathbf{a} = c\mathbf{a} + d\mathbf{a}

The mirror of the law above — spreading over a sum of scalars rather than of vectors.

Unconditional

Requires: any scalars $c, d$

The mirror of the law above — spreading over a sum of scalars rather than of vectors. Both are needed, and neither follows from the other.

viii

Multiplicative identity§ 5

1\mathbf{a} = \mathbf{a}

The one that looks like it need not be stated.

Unconditional

Requires: $1$ the scalar identity
Common error: dismissing this as trivial — without it, defining $c\mathbf{a} = \mathbf{0}$ for every $c$ satisfies all seven laws above

The one that looks like it need not be stated. It is what ties the scalars to the vectors: drop it and scaling could collapse everything to $\mathbf{0}$ while every other law still held.

Unconditional

Commutativity of addition§ 2Unconditional

\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}

Requires: component-wise, so it inherits from $\mathbb{R}$

Order of addition does not matter. Geometrically, the parallelogram is the same figure traversed two ways — tip-to-tail in either order lands on the same diagonal.

Read the full section

Associativity of addition§ 2Unconditional

(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})

Requires: any three vectors in the same $\mathbb{R}^n$

Grouping does not matter, so a chain of additions needs no parentheses. This is what lets linear combinations be written as a single sum.

Read the full section

Additive identity§ 2Unconditional

\mathbf{a} + \mathbf{0} = \mathbf{a}

Requires: $\mathbf{0}$ the vector of all zeros

The zero vector leaves everything unchanged. It is the origin — the one point every subspace must contain, since without it there is no identity to add.

Read the full section

Additive inverse§ 2Unconditional

\mathbf{a} + (-\mathbf{a}) = \mathbf{0}

Requires: one inverse per vector

Every vector can be undone by the arrow pointing the opposite way. This is what makes subtraction definable at all —

\mathbf{a} - \mathbf{b}

is shorthand for

\mathbf{a} + (-\mathbf{b})

, not a separate operation.

Read the full section

Associativity with scalars§ 5Unconditional

c(d\mathbf{a}) = (cd)\mathbf{a}

Requires: any scalars $c, d$

Scaling twice equals scaling once by the product. Note the two multiplications are different operations —

cd

happens between numbers,

c(d\mathbf{a})

between a number and a vector — and the law is the claim that they agree.

Read the full section

Distributivity over vector addition§ 5Unconditional

c(\mathbf{a} + \mathbf{b}) = c\mathbf{a} + c\mathbf{b}

Requires: any scalar $c$

Scaling spreads over a sum of vectors. Together with the next law this is precisely what linearity demands of a transformation — the definition is these two conditions and nothing more.

Read the full section

Distributivity over scalar addition§ 5Unconditional

(c + d)\mathbf{a} = c\mathbf{a} + d\mathbf{a}

Requires: any scalars $c, d$

The mirror of the law above — spreading over a sum of scalars rather than of vectors. Both are needed, and neither follows from the other.

Read the full section

Multiplicative identity§ 5Unconditional

1\mathbf{a} = \mathbf{a}

Requires: $1$ the scalar identity
Common error: dismissing this as trivial — without it, defining $c\mathbf{a} = \mathbf{0}$ for every $c$ satisfies all seven laws above

The one that looks like it need not be stated. It is what ties the scalars to the vectors: drop it and scaling could collapse everything to

\mathbf{0}

while every other law still held.

Read the full section

Every law here is unconditional in

\mathbb{R}^n

, which is what makes the space well behaved and also what makes it unremarkable. The interesting cases are elsewhere: matrix multiplication breaks commutativity and cancellation, and the dot product breaks closure. Seeing what a space looks like when nothing fails is what makes those failures legible.

The eight laws the operations obey·/linear-algebra/vectors/basic-operationsLearn Math Class

Operations at a Glance

The page treated addition, subtraction, and scalar multiplication one at a time, presenting each through three lenses — a component formula, a geometric picture, and a list of algebraic rules that the operation satisfies. The table below collects all three operations across those three views in a single place, making explicit the parallel structure that runs through the page and showing where each operation borrows from or extends the others.

Operation	Component formula	Geometric picture	Algebraic rules
Addition	a + b = (a₁ + b₁, …, aₙ + bₙ)	tip-to-tail chain or parallelogram diagonal	commutative, associative; identity 0; inverse −a
Subtraction	a − b = (a₁ − b₁, …, aₙ − bₙ)	arrow from tip of b to tip of a (common tail)	defined as a + (−b); inherits all addition rules
Scalar multiplication	ca = (ca₁, …, caₙ)	scales length by \|c\|; reverses direction if c < 0	associative with scalars; distributive (both ways); identity 1

Vector Operations FAQ

Does $(3, 4)$ mean a point or a vector?

+

Both, and only context decides. The same pair of glyphs names the location

(3,4)

in the plane and the displacement from the origin to it. Nothing in the notation distinguishes a position from an arrow, which is why the surrounding sentence has to say which reading is intended.Read more →

Can you divide one vector by another?

+

No. There is no such operation, so

\frac{\mathbf{a}}{\mathbf{b}}

is meaningless rather than merely difficult. This is also why the bold marking matters:

v

and

\mathbf{v}

obey different algebra, and dropping the mark partway through a computation is the most common writing error in the subject.Read more →

Is $\mathbb{R}^3$ the real numbers raised to a power?

+

No. The superscript counts coordinates rather than repeated multiplication:

\mathbb{R}^3

is the set of all triples of real numbers, formed by gluing three copies of

\mathbb{R}

together. Nothing is being multiplied, since there is no product of a number system by itself in the ordinary sense.Read more →

Basic Operations on Vectors

The Three Operations That Define Vector Algebra

Vector Addition

Vector Notation

Vector Notation

Properties of Addition

Commutativity

Associativity

Additive Identity

Additive Inverse

Vector Subtraction

Scalar Multiplication

Properties of Scalar Multiplication

Associativity with Scalars

Distributivity over Vector Addition

Distributivity over Scalar Addition

Multiplicative Identity

Consequences

The Algebraic Foundation

The eight laws the operations obey

Operations at a Glance

Vector Operations FAQ

Does $(3, 4)$ mean a point or a vector?

Can you divide one vector by another?

Is $\mathbb{R}^3$ the real numbers raised to a power?

Basic Operations on Vectors

The Three Operations That Define Vector Algebra

Vector Addition

Vector Notation

Vector Notation

Properties of Addition

Commutativity

Associativity

Additive Identity

Additive Inverse

Vector Subtraction

Scalar Multiplication

Properties of Scalar Multiplication

Associativity with Scalars

Distributivity over Vector Addition

Distributivity over Scalar Addition

Multiplicative Identity

Consequences

The Algebraic Foundation

The eight laws the operations obey

Operations at a Glance

Vector Operations FAQ

Does (3,4)(3, 4)(3,4) mean a point or a vector?

Can you divide one vector by another?

Is R3\mathbb{R}^3R3 the real numbers raised to a power?

Does $(3, 4)$ mean a point or a vector?

Is $\mathbb{R}^3$ the real numbers raised to a power?