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Basic Operations on Vectors



Vector Addition
Vector Notation
Properties of Addition
Vector Subtraction
Scalar Multiplication
Properties of Scalar Multiplication
The Algebraic Foundation
Operations at a Glance
Common Pitfalls
Vector Operations FAQ



The Three Operations That Define Vector Algebra

Addition, subtraction, and scalar multiplication are the operations that give vectors their algebraic structure. Each works component by component, each preserves the dimension of the input, and each carries a geometric interpretation that reinforces the computation. Together, they satisfy a precise set of rules — commutativity, associativity, distributivity — that make vector algebra predictable and consistent. Every other operation in this section, from the dot product to linear combinations, is built on top of these three.

For step-by-step results on your own numbers, use the vector addition and subtraction calculator on CalculateMatrix.




Vector Addition

Walk 4 east and 1 north, then 1 east and 3 north. Where do you end up?
Two ways of answering offer themselves: follow the arrows on a map, or add the numbers.
Following the arrows and adding the components give the same point, (5, 4).
eastnorth
first walk a41
second walk b13
end point, arrows placed tip to tail54
components added, a + b4 + 1 = 51 + 3 = 4
Each component of the end point depends only on the same component of the two walks: the east parts add, and the north parts add.
Adding vectors means adding them component by component; the picture is placing the arrows tip to tail.
Geometrically, vector addition has two equivalent visualizations. In the tip-to-tail method, the tail of b\mathbf{b} is placed at the head of a\mathbf{a}, and the sum a+b\mathbf{a} + \mathbf{b} is the vector running from the tail of a\mathbf{a} to the head of b\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 a\mathbf{a} and b\mathbf{b} visually explicit.
The definition
The sum of two vectors a\mathbf{a} and b\mathbf{b} in Rn\mathbb{R}^n is the vector whose components are the sums of their components: a+b=(a1+b1,…,an+bn)\mathbf{a} + \mathbf{b} = (a_1 + b_1, \ldots, a_n + b_n).
Adding two vectors means pairing up their components and summing each pair independently. For a=(a1,a2,…,an)\mathbf{a} = (a_1, a_2, \ldots, a_n) and b=(b1,b2,…,bn)\mathbf{b} = (b_1, b_2, \ldots, b_n) in the same Rn\mathbb{R}^n:
Vector Addition
a+b=(a1+b1, a2+b2, …, an+bn)\mathbf{a} + \mathbf{b} = (a_1 + b_1,\ a_2 + b_2,\ \ldots,\ a_n + b_n)
Learn more about this formula: Vector Addition →
So (4,1)+(1,3)=(5,4)(4, 1) + (1, 3) = (5, 4), the end point of the two walks.
Addition stays inside one space and never mixes the components.
The result is again a vector in Rn\mathbb{R}^n — addition does not change the dimension. Both inputs must belong to the same space; adding a vector in R2\mathbb{R}^2 to a vector in R3\mathbb{R}^3 is undefined because there is no way to match up the components.
u1×4u1,1u1,2u1,3u1,4+v1×4v1,1v1,2v1,3v1,4=w1×4u1+v1u2+v2u3+v3u4+v4
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.
The sum is the diagonal from the shared tail, not the line joining the tips.
Pitfall: Taking the diagonal between the tips
Reading the arrow that joins the two tips as the sum gives, for a=(4,1)\mathbf{a} = (4, 1) and b=(1,3)\mathbf{b} = (1, 3), the vector ±(3,−2)\pm(3, -2), which is a difference. The sum is the diagonal from the shared tail: a+b=(5,4)\mathbf{a} + \mathbf{b} = (5, 4).
Both constructions start from the same two arrows. The parallelogram method in particular puts several segments on the page at once: the two sides drawn from the shared tail, their two translated copies, and two diagonals that cross in the middle. The component formula above is what ties the picture to the algebra, since every segment in the figure can be written in terms of the components of a\mathbf{a} and b\mathbf{b}.
The rest of the page covers notation, the properties of addition, subtraction, scalar multiplication and its properties, the algebraic foundation, an overview of the operations and common pitfalls.

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.
v\mathbf{v} · v⃗\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: 0\mathbf{0} in Properties of Addition below is a vector, while plain 00 is a number — one equation can legally hold both, and only the typeface tells them apart.
Also writtenUnderlined v‾\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 — vv and v\mathbf{v} answer to different algebra, and ab\frac{\mathbf{a}}{\mathbf{b}} is meaningless: vectors do not divide.
v=(v1,…,vn)\mathbf{v} = (v_1, \ldots, v_n)
v with components v-one through v-n
The subscript names a coordinate slot — v2v_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 (v1,…,vn)(v_1, \ldots, v_n) and the column [v1⋮vn]\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⟨v1,v2⟩\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)(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.
Rn\mathbb{R}^n
R n — real n-space
Blackboard-bold R\mathbb{R} for the real numbers, superscript nn for how many coordinates: Rn\mathbb{R}^n is the space of all nn-component real vectors. The membership statement v∈R3\mathbf{v} \in \mathbb{R}^3 is the standard way to declare a vector's size.
CasesR2\mathbb{R}^2 is the plane, R3\mathbb{R}^3 ordinary space. Operations require matching spaces — the same-dimension rule in Vector Addition above is written as “both in Rn\mathbb{R}^n”.
Also writtenCn\mathbb{C}^n for complex components; the blackboard-bold letter family itself (N,Z,Q,R,C\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 — nn copies of R\mathbb{R} glued into tuples — not R\mathbb{R} multiplied by itself; there is nothing to multiply.
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Notation · Cheat sheet
Vector Notation at a Glance
The marks from the notation cards across the vector pages: how each is read and what it means. Every group links to the full card on its page.
Vectors
v,  v→\mathbf{v}, \; \overset{\to}{v}
vector v
Bold in print, an arrow by hand: either marks a vector, not a number.
AB⟶\overset{\longrightarrow}{AB}
vector A B
The vector from point AA to point BB.
i,j,k\mathbf{i}, \mathbf{j}, \mathbf{k}
the standard basis
Unit vectors along the axes; also e1,…,en\mathbf{e}_1, \ldots, \mathbf{e}_n.
Basic operationsThis page
(v1,…,vn)(v_1, \ldots, v_n)
v with components v one to v n
The subscript names a slot: v2v_2 is the second component, a number.
Rn\mathbb{R}^n
R n, real n-space
All vectors with nn real components.
Magnitude
∥v∥\|\mathbf{v}\|
the norm of v
Length; two bars for a vector, one for a number.
∥v∥1,  ∥v∥∞\|\mathbf{v}\|_1, \; \|\mathbf{v}\|_\infty
one-norm, infinity-norm
The subscript picks the norm; ∥v∥2\|\mathbf{v}\|_2 is the usual length.
u^\hat{\mathbf{u}}
u hat
A unit vector: ∥u^∥=1\|\hat{\mathbf{u}}\| = 1.
Dot product
a⋅b\mathbf{a} \cdot \mathbf{b}
a dot b
A number from two vectors; ab\mathbf{a}\mathbf{b} with no dot means nothing.
aTb\mathbf{a}^{T}\mathbf{b}
a transpose b
The same number in matrix form: row times column.
Cross product
a×b\mathbf{a} \times \mathbf{b}
a cross b
A vector from two vectors, in R3\mathbb{R}^3 only.
a⋅(b×c)\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})
a dot, b cross c
The scalar triple product: a signed volume.
Vector properties
a∥b\mathbf{a} \parallel \mathbf{b}
a is parallel to b
One is a scalar multiple of the other: a=cb\mathbf{a} = c\mathbf{b}.
a⊥b\mathbf{a} \perp \mathbf{b}
a is orthogonal to b
Their dot product is 00.
0\mathbf{0}
the zero vector
Parallel and orthogonal to every vector.
Same marks, different job
∣x∣|x| vs ∥v∥\|\mathbf{v}\|One pair of bars for a number, two for the length of a vector.
⋅\cdot and ×\timesBetween numbers, plain multiplication; between vectors, the dot and cross products.
v2v_2 vs v2\mathbf{v}_2Plain: the second component of v\mathbf{v}; bold: the second vector in a list.
Vector symbols tableVectors
Vector Notation at a Glance·/linear-algebra/vectorsLearn Math Class

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


Properties of Addition
a+b=b+a\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: ai+bi=bi+aia_i + b_i = b_i + a_i for every component.

Associativity


Properties of Addition
(a+b)+c=a+(b+c)(\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 a\mathbf{a}, b\mathbf{b}, and c\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


Properties of Addition
a+0=a\mathbf{a} + \mathbf{0} = \mathbf{a}

The zero vector 0=(0,0,…,0)\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


Properties of Addition
a+(−a)=0\mathbf{a} + (-\mathbf{a}) = \mathbf{0}

Every vector a\mathbf{a} has a corresponding vector −a=(−a1,−a2,…,−an)-\mathbf{a} = (-a_1, -a_2, \ldots, -a_n) that cancels it exactly. The inverse has the same magnitude as a\mathbf{a} but points in the opposite direction. Adding a vector to its inverse returns to the origin — the displacements undo each other completely.
a + b and b + a: two routes to the same diagonalOabbaa + b = b + adashed: the same vector, moved to the other one’s tipa then b, or b then a: the same diagonal
Commutativity, drawn

Walk a first and then b (lower route), or b first and then a (upper route): both walks end at the same point, the far corner of the parallelogram that a and b span. The amber diagonal from O to that corner is a + b, and it is also b + a. The dashed arrows are not new vectors — they are a and b moved so that each starts where the other ends.

Because the two routes always meet at the same corner, a sum of vectors can be taken in any order.
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 a−b\mathbf{a} - \mathbf{b} is defined as:

Vector Subtraction
a−b=a+(−b)=(a1−b1, a2−b2, …, an−bn)\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 →


Reducing subtraction to addition and negation means that no separate set of rules is needed: differences are handled by the four properties of addition. Subtraction itself is neither commutative nor associative — a−b=−(b−a)\mathbf{a} - \mathbf{b} = -(\mathbf{b} - \mathbf{a}).

Pitfall: Reversing the order of subtraction
Drawing a−b\mathbf{a} - \mathbf{b} from the tip of a\mathbf{a} to the tip of b\mathbf{b} gives, for a=(5,2)\mathbf{a} = (5, 2) and b=(2,4)\mathbf{b} = (2, 4), the vector (−3,2)(-3, 2), which is b−a\mathbf{b} - \mathbf{a}. The difference is (5−2, 2−4)=(3,−2)(5 - 2,\ 2 - 4) = (3, -2).

The geometric picture of subtraction is particularly useful. When a\mathbf{a} and b\mathbf{b} are drawn from a common tail, the difference a−b\mathbf{a} - \mathbf{b} is the vector pointing from the tip of b\mathbf{b} to the tip of a\mathbf{a}. This interpretation connects subtraction directly to distance: the length of a−b\mathbf{a} - \mathbf{b} is the Euclidean distance between the heads of the two vectors, formalized as d(a,b)=∥a−b∥d(\mathbf{a}, \mathbf{b}) = \|\mathbf{a} - \mathbf{b}\| on the magnitude page.
a − b runs from the tip of b to the tip of aOa = (5, 2)b = (2, 4)a − b = (3, −2)b followed by a − b lands on a, so b + (a − b) = aits length, √13, is the distance between the two tipsa − b runs from the tip of b to the tip of a
Subtraction as the arrow between two tips

With a and b drawn from the same point, a − b is the amber arrow from the tip of b to the tip of a. Check it tip-to-tail: walk b, then a − b, and you arrive at the tip of a, so b + (a − b) = a. Here a − b = (5 − 2, 2 − 4) = (3, −2), and its length √13 is exactly how far apart the two tips are.

Reading a − b as the arrow between two tips is what turns subtraction into a way of measuring distance.

Scalar Multiplication

Scalar multiplication takes a real number cc and a vector a\mathbf{a} and scales every component by cc:

Scalar Multiplication of Vectors
ca=(ca1, ca2, …, can)c\mathbf{a} = (ca_1,\ ca_2,\ \ldots,\ ca_n)
Learn more about this formula: Scalar Multiplication of Vectors →


The result is always a vector in the same Rn\mathbb{R}^n.

RULE
Geometrically, scalar multiplication changes the length of the vector by a factor of ∣c∣|c| while preserving or reversing its direction depending on the sign of cc.

Pitfall: Giving a negative scalar a negative length
Writing ∥−2a∥=−2∥a∥\|-2\mathbf{a}\| = -2\|\mathbf{a}\| for a=(3,4)\mathbf{a} = (3, 4) gives a length of −10-10. But −2a=(−6,−8)-2\mathbf{a} = (-6, -8) has length 36+64=10=∣−2∣⋅5\sqrt{36 + 64} = 10 = |-2| \cdot 5; the sign only reverses the direction.

When c>0c > 0, the scaled vector cac\mathbf{a} points in the same direction as a\mathbf{a}. If c>1c > 1, the vector stretches; if 0<c<10 < c < 1, it compresses. When c<0c < 0, the direction flips — the scaled vector points opposite to a\mathbf{a}, with its length multiplied by ∣c∣|c|. The boundary case c=0c = 0 collapses any vector to the zero vector: 0a=00\mathbf{a} = \mathbf{0}.

Because scalar multiplication only ever stretches, compresses, or reverses a vector along its own line, the result cac\mathbf{a} is always parallel to a\mathbf{a} (provided a≠0\mathbf{a} \neq \mathbf{0}). This observation becomes important later: two vectors are parallel precisely when one is a scalar multiple of the other.

Fact: Every quadrilateral hides a parallelogram
Joining the midpoints of the four sides of any quadrilateral, even a non-convex one, gives a parallelogram. With corners a,b,c,d\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d} in order, the midpoints 12(a+b)\frac{1}{2}(\mathbf{a} + \mathbf{b}) and 12(b+c)\frac{1}{2}(\mathbf{b} + \mathbf{c}) differ by 12(c−a)\frac{1}{2}(\mathbf{c} - \mathbf{a}), and so do the midpoints of the other two sides.

Scaling by 12\frac{1}{2} also locates midpoints. The point halfway between the tips of a\mathbf{a} and b\mathbf{b} is a+12(b−a)=12(a+b)\mathbf{a} + \frac{1}{2}(\mathbf{b} - \mathbf{a}) = \frac{1}{2}(\mathbf{a} + \mathbf{b}), the average of the two vectors. The arrow from one midpoint to another is then a difference of two such averages, and comparing those arrows component by component shows whether two segments are parallel and equally long, with no angles or rulers involved.
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
k·u1×4u1,1u1,2u1,3u1,4=w1×4k·u1k·u2k·u3k·u4
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


Properties of Scalar Multiplication
c(da)=(cd)ac(d\mathbf{a}) = (cd)\mathbf{a}

Scaling a vector by dd and then by cc produces the same result as scaling once by the product cdcd. The operations collapse into a single multiplication on the scalar side.

Distributivity over Vector Addition


Properties of Scalar Multiplication
c(a+b)=ca+cbc(\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 cc yields the same vector as scaling both sides by cc and then forming the new parallelogram.

Distributivity over Scalar Addition


Properties of Scalar Multiplication
(c+d)a=ca+da(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


Properties of Scalar Multiplication
1a=a1\mathbf{a} = \mathbf{a}

Scaling by 11 leaves a vector unchanged. This ensures that the scalar 11 acts neutrally, just as the zero vector acts neutrally under addition.

Consequences


Two special cases follow immediately from these rules. Setting c=0c = 0 gives 0a=00\mathbf{a} = \mathbf{0}: scaling any vector by zero produces the zero vector. Setting c=−1c = -1 gives (−1)a=−a(-1)\mathbf{a} = -\mathbf{a}: scaling by −1-1 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 c1v1+c2v2+⋯+ckvkc_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.

Fact: One law follows from the other seven
Expanding (1+1)(a+b)(1 + 1)(\mathbf{a} + \mathbf{b}) by each distributive law gives a+a+b+b=a+b+a+b\mathbf{a} + \mathbf{a} + \mathbf{b} + \mathbf{b} = \mathbf{a} + \mathbf{b} + \mathbf{a} + \mathbf{b}. Cancelling the first a\mathbf{a} and the last b\mathbf{b} on both sides leaves a+b=b+a\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}, so commutativity never has to be assumed.

The eight properties listed on this page — four for addition and four for scalar multiplication, two of them distributive laws — are not arbitrary. Together with closure, 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 R3\mathbb{R}^3, polynomials, matrices, or functions. The vectors in Rn\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§ 2a+b=b+a\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}
Order of addition does not matter.
Unconditional
Requires
component-wise, so it inherits from R\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(a+b)+c=a+(b+c)(\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 Rn\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§ 2a+0=a\mathbf{a} + \mathbf{0} = \mathbf{a}
The zero vector leaves everything unchanged.
Unconditional
Requires
0\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§ 2a+(−a)=0\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 — a−b\mathbf{a} - \mathbf{b} is shorthand for a+(−b)\mathbf{a} + (-\mathbf{b}), not a separate operation.

v
Associativity with scalars§ 5c(da)=(cd)ac(d\mathbf{a}) = (cd)\mathbf{a}
Scaling twice equals scaling once by the product.
Unconditional
Requires
any scalars c,dc, d

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

vi
Distributivity over vector addition§ 5c(a+b)=ca+cbc(\mathbf{a} + \mathbf{b}) = c\mathbf{a} + c\mathbf{b}
Scaling spreads over a sum of vectors.
Unconditional
Requires
any scalar cc

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)a=ca+da(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,dc, 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§ 51a=a1\mathbf{a} = \mathbf{a}
The one that looks like it need not be stated.
Unconditional
Requires
11 the scalar identity
Common error
dismissing this as trivial — without it, defining ca=0c\mathbf{a} = \mathbf{0} for every cc 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 0\mathbf{0} while every other law still held.

Unconditional
Commutativity of addition§ 2Unconditional
a+b=b+a\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}
Requires
component-wise, so it inherits from R\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
(a+b)+c=a+(b+c)(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})
Requires
any three vectors in the same Rn\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
a+0=a\mathbf{a} + \mathbf{0} = \mathbf{a}
Requires
0\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
a+(−a)=0\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 — a−b\mathbf{a} - \mathbf{b} is shorthand for a+(−b)\mathbf{a} + (-\mathbf{b}), not a separate operation.
Read the full section
Associativity with scalars§ 5Unconditional
c(da)=(cd)ac(d\mathbf{a}) = (cd)\mathbf{a}
Requires
any scalars c,dc, d
Scaling twice equals scaling once by the product. Note the two multiplications are different operations — cdcd happens between numbers, c(da)c(d\mathbf{a}) between a number and a vector — and the law is the claim that they agree.
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Distributivity over vector addition§ 5Unconditional
c(a+b)=ca+cbc(\mathbf{a} + \mathbf{b}) = c\mathbf{a} + c\mathbf{b}
Requires
any scalar cc
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.
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Distributivity over scalar addition§ 5Unconditional
(c+d)a=ca+da(c + d)\mathbf{a} = c\mathbf{a} + d\mathbf{a}
Requires
any scalars c,dc, 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.
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Multiplicative identity§ 5Unconditional
1a=a1\mathbf{a} = \mathbf{a}
Requires
11 the scalar identity
Common error
dismissing this as trivial — without it, defining ca=0c\mathbf{a} = \mathbf{0} for every cc 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 0\mathbf{0} while every other law still held.
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Every law here is unconditional in Rn\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
Anything that can be added and scaled under these eight laws is a vector, whether an arrow, a polynomial or a function.

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

Common Pitfalls

Three mistakes account for most wrong answers with the basic operations. Each one is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.

Taking the diagonal between the tips

In the parallelogram method, a\mathbf{a} and b\mathbf{b} are drawn from a shared tail, and the arrow joining their two tips is read off as the sum. That arrow is a diagonal of the same parallelogram, but it is the wrong one.

Wrong working: for a=(4,1)\mathbf{a} = (4, 1) and b=(1,3)\mathbf{b} = (1, 3), the arrow from the tip of b\mathbf{b} to the tip of a\mathbf{a} is
(4−1, 1−3)=(3,−2)(4 - 1,\ 1 - 3) = (3, -2)

and this is taken as a+b\mathbf{a} + \mathbf{b}.

Right working: the sum is the diagonal that starts at the shared tail and ends at the opposite corner.
a+b=(4+1, 1+3)=(5,4)\mathbf{a} + \mathbf{b} = (4 + 1,\ 1 + 3) = (5, 4)

The arrow between the tips is a−b=(3,−2)\mathbf{a} - \mathbf{b} = (3, -2), the difference.

Why it happens: both diagonals are drawn from the same two arrows, and the one joining the tips looks like the natural link between a\mathbf{a} and b\mathbf{b}.

How to check: build the sum tip-to-tail. Walking a\mathbf{a} from the origin ends at (4,1)(4, 1), and walking b\mathbf{b} from there ends at (5,4)(5, 4), so the sum must reach (5,4)(5, 4). The lengths also disagree: ∥(3,−2)∥=13≈3.61\|(3, -2)\| = \sqrt{13} \approx 3.61, while ∥(5,4)∥=41≈6.40\|(5, 4)\| = \sqrt{41} \approx 6.40.

Reversing the order of subtraction

The difference a−b\mathbf{a} - \mathbf{b} is drawn from the tip of a\mathbf{a} to the tip of b\mathbf{b}, or computed by subtracting the components of a\mathbf{a} from those of b\mathbf{b}. Both give b−a\mathbf{b} - \mathbf{a}, the opposite vector.

Wrong working: for a=(5,2)\mathbf{a} = (5, 2) and b=(2,4)\mathbf{b} = (2, 4),
a−b=(2−5, 4−2)=(−3,2)\mathbf{a} - \mathbf{b} = (2 - 5,\ 4 - 2) = (-3, 2)


Right working: subtract the components of b\mathbf{b} from those of a\mathbf{a}.
a−b=(5−2, 2−4)=(3,−2)\mathbf{a} - \mathbf{b} = (5 - 2,\ 2 - 4) = (3, -2)

Drawn from a common tail, this is the arrow from the tip of b\mathbf{b} to the tip of a\mathbf{a}.

Why it happens: the expression is read left to right, "from a\mathbf{a} to b\mathbf{b}", while the arrow for a−b\mathbf{a} - \mathbf{b} ends at a\mathbf{a}, the first vector written.

How to check: add b\mathbf{b} back. The result must be a\mathbf{a}: (2,4)+(3,−2)=(5,2)(2, 4) + (3, -2) = (5, 2), as it should. The wrong answer gives (2,4)+(−3,2)=(−1,6)(2, 4) + (-3, 2) = (-1, 6), which is not a\mathbf{a}.

Giving a negative scalar a negative length

The length of cac\mathbf{a} is found by multiplying the length of a\mathbf{a} by cc itself, so a negative scalar produces a negative length.

Wrong working: for a=(3,4)\mathbf{a} = (3, 4), ∥a∥=5\|\mathbf{a}\| = 5, and
∥−2a∥=−2⋅5=−10\|-2\mathbf{a}\| = -2 \cdot 5 = -10


Right working: the length is multiplied by ∣c∣|c|.
−2a=(−6,−8),∥−2a∥=36+64=10=∣−2∣⋅5-2\mathbf{a} = (-6, -8), \qquad \|-2\mathbf{a}\| = \sqrt{36 + 64} = 10 = |-2| \cdot 5

The minus sign does not shrink anything; it turns a\mathbf{a} around to point the opposite way.

Why it happens: the rule ∥ca∥=c∥a∥\|c\mathbf{a}\| = c\|\mathbf{a}\| gives the right answer for every positive cc, so the absolute value gets left out.

How to check: a length is never negative. Compute the components of cac\mathbf{a} and take their length directly; the result must equal ∣c∣|c| times ∥a∥\|\mathbf{a}\|.

Vector Operations FAQ

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

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Both, and only context decides. The same pair of glyphs names the location (3,4)(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?

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No. There is no such operation, so ab\frac{\mathbf{a}}{\mathbf{b}} is meaningless rather than merely difficult. This is also why the bold marking matters: vv and v\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 R3\mathbb{R}^3 the real numbers raised to a power?

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No. The superscript counts coordinates rather than repeated multiplication: R3\mathbb{R}^3 is the set of all triples of real numbers, formed by gluing three copies of R\mathbb{R} together. Nothing is being multiplied, since there is no product of a number system by itself in the ordinary sense.Read more →

Which diagonal of the parallelogram is a+b\mathbf{a} + \mathbf{b}?

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The one that starts at the shared tail and runs to the opposite corner. The other diagonal, joining the two tips, is a difference: drawn from the tip of b\mathbf{b} to the tip of a\mathbf{a} it is a−b\mathbf{a} - \mathbf{b}. For a=(4,1)\mathbf{a} = (4, 1) and b=(1,3)\mathbf{b} = (1, 3) the sum is (5,4)(5, 4), while the arrow between the tips is ±(3,−2)\pm(3, -2). A tip-to-tail walk settles it: a\mathbf{a} and then b\mathbf{b} ends at (5,4)(5, 4).Read more →

How do you find the midpoint between the tips of two vectors?

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Average them: the midpoint is 12(a+b)\frac{1}{2}(\mathbf{a} + \mathbf{b}). Walk to the tip of a\mathbf{a}, then halfway along b−a\mathbf{b} - \mathbf{a}, and a+12(b−a)=12(a+b)\mathbf{a} + \frac{1}{2}(\mathbf{b} - \mathbf{a}) = \frac{1}{2}(\mathbf{a} + \mathbf{b}). The same move proves geometry without angles: joining the midpoints of the four sides of any quadrilateral a,b,c,d\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d} gives a parallelogram, because two opposite sides of the inner figure both equal 12(c−a)\frac{1}{2}(\mathbf{c} - \mathbf{a}).Read more →

Does commutativity of vector addition have to be assumed?

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No, it follows from the other seven laws. Expand (1+1)(a+b)(1 + 1)(\mathbf{a} + \mathbf{b}) two ways: distributing the sum of scalars first gives a+b+a+b\mathbf{a} + \mathbf{b} + \mathbf{a} + \mathbf{b}, distributing over the sum of vectors first gives a+a+b+b\mathbf{a} + \mathbf{a} + \mathbf{b} + \mathbf{b}. Cancelling the first a\mathbf{a} and the last b\mathbf{b} on both sides, with inverses and associativity, leaves a+b=b+a\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}.Read more →