Vector Properties: Magnitude, Direction & Orthogonality

Magnitude

The arrow from the origin to (3, 4) is 5 long — the hypotenuse of a 3-4-5 right triangle.

Pythagoras measures arrows in the plane. Does the same recipe work with more coordinates?

Applied once more for each extra coordinate, Pythagoras gives every length.

vector	space	square, add, take the root	length
(3, 4)	R²	√(9 + 16)	5
(1, 2, 2)	R³	√(1 + 4 + 4)	3
(1, 1, 1, 1)	R⁴	√(1 + 1 + 1 + 1)	2

In

\mathbb{R}^4

there is no picture, but the recipe still gives a number.

Square the components, add, take the root: one recipe measures a vector in any dimension.

Each extra coordinate is one more leg at right angles to all the others, so it adds one more square under the root.

The definition

The magnitude of a vector

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

is its length,

\|\mathbf{v}\| = \sqrt{v_1^2 + \cdots + v_n^2}

.

Every vector has a magnitude — a single non-negative number that measures its size. For a vector

\mathbf{v} = (v_1, v_2, \ldots, v_n)

in

\mathbb{R}^n

, the magnitude is given by the Euclidean norm:

Magnitude

\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}

So

\|(3, 4)\| = 5

,

\|(1, 2, 2)\| = 3

and

\|(1, 1, 1, 1)\| = 2

.

Magnitude belongs to the vector alone: a non-negative number, zero only for the zero vector.

Magnitude is an intrinsic property: it belongs to the vector and does not depend on any other vector or operation. It is always non-negative, and the only vector with magnitude zero is the zero vector

\mathbf{0}

. This makes magnitude a reliable indicator of whether a vector is trivial — a vector is the zero vector if and only if its magnitude vanishes.

Lengths do not simply add.

The norm satisfies two additional structural properties. Under scalar multiplication, the magnitude scales predictably:

\|c\mathbf{v}\| = |c|\,\|\mathbf{v}\|

. Under addition, it obeys the triangle inequality:

\|\mathbf{a} + \mathbf{b}\| \leq \|\mathbf{a}\| + \|\mathbf{b}\|

. These properties, along with normalization and the distance formula, are developed on the magnitude page.

For

(3, 0)

and

(0, 4)

the lengths are

3

and

4

, but their sum

(3, 4)

has length

5

, not

7

.

The rest of the page covers direction, dimension, notation, equality, parallelism, orthogonality, the algebraic properties, a summary and common pitfalls.

Direction

Every nonzero vector has a direction — an orientation in space that specifies which way the vector points. Direction is what separates a vector from a scalar: a scalar carries size alone, while a vector carries size together with a spatial orientation.

In

\mathbb{R}^2

, direction can be described by the angle

\alpha

that the vector makes with the positive

x

-axis. A vector

(v_1, v_2)

points in the direction

\alpha = \arctan\left(\frac{v_2}{v_1}\right)

, with appropriate adjustment for quadrant. In

\mathbb{R}^3

, a single angle is insufficient — direction is instead specified by two angles or, more commonly, by the unit vector

\frac{\mathbf{v}}{\|\mathbf{v}\|}

that points the same way with magnitude stripped away.

Pitfall: Skipping the quadrant adjustment

Reading the angle of

(-1, -1)

as

\arctan\left(\frac{-1}{-1}\right) = \arctan(1) = 45^\circ

points it into the first quadrant. Both components are negative, so the vector lies in the third quadrant:

\alpha = 45^\circ + 180^\circ = 225^\circ

.

Finding the angle of a specific vector takes two readings of its components. The ratio

\frac{v_2}{v_1}

fixes the slope of the line the vector lies along, and the signs of

v_1

and

v_2

fix which way along that line it points. Both readings go into the final angle, measured from the positive

x

-axis.

The unit vector representation generalizes to any

\mathbb{R}^n

. Dividing a nonzero vector by its magnitude isolates its direction as a vector on the unit sphere. Two vectors that differ only in magnitude share the same unit vector and therefore point the same way.

The zero vector has no direction. Its components are all zero, its magnitude is zero, and there is no meaningful orientation to extract. The expression

\frac{\mathbf{0}}{\|\mathbf{0}\|}

involves division by zero and is undefined. This is not a technicality — the zero vector genuinely has no directional content.

Direction is what remains of a vector once its length is divided out, so the zero vector has none.

Dimension

Every vector belongs to a specific

\mathbb{R}^n

, and the number

n

— the count of components — is its dimension. A vector in

\mathbb{R}^2

has two components, a vector in

\mathbb{R}^5

has five, and the distinction is absolute: there is no natural way to add or compare vectors from different dimensions because there is no way to match up their components.

In

\mathbb{R}^2

and

\mathbb{R}^3

, dimension has a direct geometric meaning — it is the number of independent spatial directions available. Two-component vectors live in a plane; three-component vectors live in the full space we inhabit visually. For

n > 3

, the geometric picture breaks down, but the algebraic structure does not. Operations, norms, and dot products all work identically in

\mathbb{R}^{100}

as they do in

\mathbb{R}^3

— only the number of terms in each sum changes.

Dimension is fixed by the space, not by the vector. Every vector in

\mathbb{R}^n

has exactly

n

components, and all operations within

\mathbb{R}^n

produce vectors with

n

components. A vector cannot gain or lose components through addition, scaling, or any linear combination. This rigidity is what keeps the algebra consistent.

Vector Property Notation

Notation

Vector Property Notation

The properties above are written one way and the properties below another, and the shape of the mark tells you which kind you are reading. Four points decide the rest.

The double bars of

\|\mathbf{v}\|

, the hat on a unit vector and the norm subscripts belong to magnitude. The dot in

\mathbf{a} \cdot \mathbf{b}

is set out at the dot product, the perpendicular sign

\perp

at inner products, and bold type, component tuples and

\mathbb{R}^n

at basic operations.

\|\mathbf{v}\| \quad\text{versus}\quad \mathbf{a} \parallel \mathbf{b}

something applied to one vector, against something written between two

The two kinds of property on this page are distinguishable before the symbols are read. An intrinsic property is written as an operation on a single vector and returns a number. A relational property is written as a sign standing between two vectors and returns a yes or no.

CasesMagnitude, direction and dimension take one argument. Equality, parallelism and orthogonality take two and assert nothing about either vector on its own — a vector is not parallel, it is parallel to something.

Also writtenRelational signs are sometimes negated with a slash, as in

\mathbf{a} \nparallel \mathbf{b}

, which has no counterpart on the intrinsic side because there is nothing there to deny.

Do not confuseTreating direction as a one-vector quantity in the way magnitude is. Direction has no single-number output; the closest thing is the unit vector, which is another vector, and the angle it makes is measured against a reference that has to be chosen.

\mathbf{a} \parallel \mathbf{b} \iff \mathbf{a} = c\mathbf{b}

a is parallel to b exactly when a is some scalar multiple of b

The sign is borrowed from elementary geometry, where it joins two lines. Here it joins two vectors, and the definition it stands for is wider than the word suggests: any scalar

c

qualifies, including negative ones, so a vector and its reverse are parallel.

CasesThe symbol records no distinction between

c > 0

and

c < 0

. The second case is called anti-parallel when the difference matters, and it needs the word because the mark cannot carry it.

Also writtenSome texts reserve

\parallel

for the positive case and write

\mathbf{a} \uparrow\downarrow \mathbf{b}

for the reverse, so the same symbol means slightly different things in different books.

Do not confuseThe double bars of the norm.

\|\mathbf{a}\|

and

\mathbf{a} \parallel \mathbf{b}

use two vertical strokes for unrelated jobs — one encloses, the other separates — and both can appear in a single line of working. Position is the only clue.

Same glyph elsewhereIn geometry the same sign relates lines and planes, where parallel objects never meet. Vectors have no position, so that reading does not transfer: parallel vectors lie along one line through the origin.

\mathbf{0} \parallel \mathbf{v} \quad\text{and}\quad \mathbf{0} \perp \mathbf{v} \quad \text{for every } \mathbf{v}

the zero vector is parallel to every vector, and orthogonal to every vector

Parallelism and orthogonality are presented as opposite extremes, and one object sits at both. Neither statement is forced by geometry — the zero vector has no direction to align or oppose. They are conventions, adopted so that theorems about parallel or orthogonal vectors need no clause excluding

\mathbf{0}

.

CasesEach convention has a computation behind it that comes out the right way:

\mathbf{0} = 0\mathbf{v}

satisfies the definition of parallel, and

\mathbf{0} \cdot \mathbf{v} = 0

satisfies the definition of orthogonal. The definitions were written in a form that admits the zero vector rather than being patched afterwards.

Also writtenStated as the zero vector is parallel and orthogonal to everything, which is accurate and sounds contradictory. The contradiction is only in the words: no vector other than

\mathbf{0}

has both relations to a given vector.

Do not confuseConcluding that

\parallel

and

\perp

are not really opposites. They are, everywhere except at the one vector that has no direction, and the overlap is a property of

\mathbf{0}

rather than of the two relations.

\frac{a_1}{b_1} = \frac{a_2}{b_2} = \cdots = \frac{a_n}{b_n}

the component ratios are all equal

A test written as a chain of equalities, each link comparing one pair of components. The chain says the ratio is the same throughout, and that shared value is the scalar

c

that the definition of parallelism asks for, recovered without being solved for.

CasesThe form is undefined exactly where it is most likely to be used. Any zero component of

\mathbf{b}

puts a zero in a denominator, and the vectors may still be parallel —

(0, 3)

and

(0, 6)

are, while the first ratio cannot be written.

Also writtenCross-multiplied as

a_ib_j = a_jb_i

for every pair

i, j

, which has no denominators and no exceptions. This is the form to use when either vector might have a zero component.

Do not confuseReading the parenthetical with appropriate handling as a minor caveat. It is where the notation stops working, and the cross-multiplied version exists because of it.

All linear algebra symbols →↑ Vectors

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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

\mathbf{v}, \; \overset{\to}{v}

vector v

Bold in print, an arrow by hand: either marks a vector, not a number.

\overset{\longrightarrow}{AB}

vector A B

The vector from point

A

to point

B

.

\mathbf{i}, \mathbf{j}, \mathbf{k}

the standard basis

Unit vectors along the axes; also

\mathbf{e}_1, \ldots, \mathbf{e}_n

.

Basic operations

(v_1, \ldots, v_n)

v with components v one to v n

The subscript names a slot:

v_2

is the second component, a number.

\mathbb{R}^n

R n, real n-space

All vectors with

n

real components.

Magnitude

\|\mathbf{v}\|

the norm of v

Length; two bars for a vector, one for a number.

\|\mathbf{v}\|_1, \; \|\mathbf{v}\|_\infty

one-norm, infinity-norm

The subscript picks the norm;

\|\mathbf{v}\|_2

is the usual length.

\hat{\mathbf{u}}

u hat

A unit vector:

\|\hat{\mathbf{u}}\| = 1

.

Dot product

\mathbf{a} \cdot \mathbf{b}

a dot b

A number from two vectors;

\mathbf{a}\mathbf{b}

with no dot means nothing.

\mathbf{a}^{T}\mathbf{b}

a transpose b

The same number in matrix form: row times column.

Cross product

\mathbf{a} \times \mathbf{b}

a cross b

A vector from two vectors, in

\mathbb{R}^3

only.

\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})

a dot, b cross c

The scalar triple product: a signed volume.

Vector propertiesThis page

\mathbf{a} \parallel \mathbf{b}

a is parallel to b

One is a scalar multiple of the other:

\mathbf{a} = c\mathbf{b}

.

\mathbf{a} \perp \mathbf{b}

a is orthogonal to b

Their dot product is

0

.

\mathbf{0}

the zero vector

Parallel and orthogonal to every vector.

Same marks, different job

|x|

vs

\|\mathbf{v}\|

One pair of bars for a number, two for the length of a vector.

\cdot

and

\times

Between numbers, plain multiplication; between vectors, the dot and cross products.

v_2

vs

\mathbf{v}_2

Plain: the second component of

\mathbf{v}

; bold: the second vector in a list.

Vector symbols table Vectors

Vector Notation at a Glance·/linear-algebra/vectorsLearn Math Class

Equality

Two vectors are equal when every one of their corresponding components matches. For

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

and

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

:

Vector Equality

\mathbf{a} = \mathbf{b} \iff a_i = b_i \text{ for every } i = 1, 2, \ldots, n

Browse all linear algebra formulas →

Pitfall: Comparing only the lengths

Calling

(3, 4)

and

(4, 3)

equal because both have length

5

skips the components. They differ in the first slot,

3 \neq 4

, so

(3, 4) \neq (4, 3)

: the two arrows point in different directions.

There is no partial equality for vectors. If even a single component differs, the vectors are not equal. This all-or-nothing criterion is the algebraic version of the geometric requirement: equal vectors must have the same magnitude and the same direction.

For free vectors — vectors defined by magnitude and direction alone, without a fixed position — equality is independent of location. An arrow drawn at one corner of a diagram represents the same vector as an identical arrow drawn elsewhere, as long as the length and orientation agree. Moving a vector without rotating or rescaling it does not change the vector. This is the principle that allows vectors to be repositioned freely in tip-to-tail constructions and parallelogram diagrams without altering the result of an addition.

Parallelism

Two vectors are parallel when one is a scalar multiple of the other. For

\mathbf{a}

and

\mathbf{b}

with

\mathbf{b} \neq \mathbf{0}

:

Parallel Vectors

\mathbf{a} \parallel \mathbf{b} \iff \mathbf{a} = c\mathbf{b} \text{ for some } c \in \mathbb{R}

Browse all linear algebra formulas →

When

c > 0

, the two vectors point in the same direction — they are parallel in the strict sense. When

c < 0

, they point in opposite directions — anti-parallel. Both cases fall under the umbrella of parallelism because the vectors lie along the same line through the origin, differing only in scale and possibly in sign.

Parallelism can be detected without computing the scalar

c

explicitly. If the components of

\mathbf{a}

and

\mathbf{b}

satisfy

\frac{a_1}{b_1} = \frac{a_2}{b_2} = \cdots = \frac{a_n}{b_n}

(with appropriate handling when a component of

\mathbf{b}

is zero), the vectors are parallel. Alternatively, in

\mathbb{R}^3

, the cross product provides a definitive test:

\mathbf{a} \times \mathbf{b} = \mathbf{0}

if and only if

\mathbf{a}

and

\mathbf{b}

are parallel.

Fact: Parallel to parallel, yet perpendicular

(1, 0)

and

(0, 1)

are each parallel to

\mathbf{0}

under the convention below, yet they meet at a right angle. Passing parallelism along a chain works only when the middle vector is nonzero.

Among nonzero vectors, parallelism passes along a chain. If

\mathbf{a} = c\mathbf{b}

and

\mathbf{b} = d\mathbf{w}

, then

\mathbf{a} = (cd)\mathbf{w}

, so a vector parallel to a vector parallel to

\mathbf{w}

is itself parallel to

\mathbf{w}

. This is what lets a whole family of vectors be described as lying along one line through the origin.

By convention, the zero vector is considered parallel to every vector, since

\mathbf{0} = 0\mathbf{b}

for any

\mathbf{b}

. This convention avoids the need to exclude

\mathbf{0}

as a special case in statements about parallelism.

Parallelism is the simplest instance of linear dependence.

THEOREM

Two vectors are linearly dependent precisely when one is a scalar multiple of the other — precisely when they are parallel.

The concept generalizes: for three or more vectors, dependence means at least one vector is a linear combination of the others, but for a pair, dependence reduces to parallelism.

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Same line, three ways

Two vectors are parallel when one is a scalar multiple of the other:

\mathbf{a} = c\mathbf{b}

. The value of

c

says how they lie along that line.

Parallel vectors

\mathbf{a} = c\mathbf{b}

split by the scalar $c$

+

Case 1

c > 0

Same direction

parallel in the strict sense

−

Case 2

c < 0

Opposite directions

anti-parallel

0

Case 3

c = 0

The zero vector

by convention

\mathbf{0} = 0\mathbf{b}

is parallel to every vector

Takeaway

For a pair of vectors, being parallel and being linearly dependent are the same thing.

Parallel vectors: the sign of c·/linear-algebra/vectors/propertiesLearn Math Class

One vector a multiple of the other

Both arrows lie along the same line, and the region their combinations can reach has collapsed onto that line with them. Parallel is the geometric word and linearly dependent is the algebraic one, but the picture is the same picture: a second vector that adds no direction the first did not already have. Drag one arrow off the line and watch the plane come back on the span and independence explorer.

This is the first appearance of an idea the vector-spaces pages return to constantly: what matters is not how many vectors you have but how many directions they supply.

Orthogonality

Two vectors are orthogonal when their dot product equals zero:

Orthogonality

\mathbf{a} \perp \mathbf{b} \quad \Longleftrightarrow \quad \mathbf{a} \cdot \mathbf{b} = 0

In

\mathbb{R}^2

and

\mathbb{R}^3

, orthogonality corresponds to perpendicularity — the two vectors meet at a right angle. The algebraic condition

a_1b_1 + a_2b_2 + \cdots + a_nb_n = 0

translates the geometric concept of a

90°

angle into a computation that works in any dimension, including those beyond visualization.

Pitfall: Reading not orthogonal as parallel

Concluding that

(1, 2)

and

(3, 1)

are parallel because

1 \cdot 3 + 2 \cdot 1 = 5 \neq 0

skips a case. Their ratios differ,

\frac{1}{3} \neq \frac{2}{1}

, so the pair is neither parallel nor orthogonal: the vectors meet at

45^\circ

.

Orthogonality is in a sense the opposite extreme of parallelism. Parallel vectors are maximally aligned — one lies entirely along the direction of the other. Orthogonal vectors have zero alignment — projecting one onto the other yields the zero vector. Between these extremes lies every other angular relationship, measured quantitatively by the dot product.

The zero vector is orthogonal to every vector, since

\mathbf{0} \cdot \mathbf{v} = 0

for all

\mathbf{v}

. This is consistent with the convention that the zero vector is also parallel to every vector. The zero vector is the only vector that is simultaneously parallel and orthogonal to everything — a consequence of its having no magnitude and no direction.

Orthogonality grows in importance well beyond this section. Orthogonal bases simplify coordinate computations, orthogonal projections decompose vectors into independent components, and the Gram–Schmidt process converts an arbitrary basis into an orthogonal one. These ideas are developed in the orthogonality section.

Relation	Algebraic condition	Geometric meaning	Computational test
Parallel a ∥ b	a = cb for some scalar c	same direction (c > 0) or opposite direction (c < 0); both lie on the same line through the origin	check component ratios; in ℝ³, cross product a × b = 0
Orthogonal a ⊥ b	a · b = 0	meet at a right angle (90°); zero alignment between the vectors	compute the dot product directly; works in any ℝⁿ

Orthogonality turns a right angle into one equation, a dot product of zero, that works in every dimension.

Algebraic Properties

In addition to the intrinsic and relational properties above, vectors in

\mathbb{R}^n

obey a collection of algebraic rules governing how they interact with addition and scalar multiplication. These rules are not properties of individual vectors but of the operations themselves — they describe the behavior of addition and scaling as applied to any vectors in the space.

Under addition, vectors are commutative (

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

), associative (

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

), have an identity element (

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

), and every vector has an additive inverse (

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

). Under scalar multiplication, associativity holds (

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

), the scalar

1

acts as an identity (

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

), and two distributive laws connect the two operations:

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

and

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

.

Pitfall: Scaling only the first vector

Writing

2\big((1, 0) + (0, 3)\big)

as

(2, 0) + (0, 3) = (2, 3)

leaves the second vector unscaled. The scalar multiplies every term:

(2, 0) + (0, 6) = (2, 6)

.

In a computation, the distributive laws let a scalar be applied before or after an addition. The vector

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

can be found by adding first and scaling the sum, or by scaling

\mathbf{a}

and

\mathbf{b}

separately and adding the results. The two routes always end at the same vector, so either one serves as a check on the other.

These eight properties are not specific to vectors in

\mathbb{R}^n

— together with closure, they are the axioms that define a vector space. Any mathematical structure satisfying the same rules qualifies as a vector space, whether its elements are arrows in the plane, polynomials, matrices, or functions. The vectors in

\mathbb{R}^n

are the most tangible example, but the algebraic framework they satisfy is far more general.

Summary: Properties at a Glance

This page has introduced six properties of vectors — three intrinsic to a single vector (magnitude, direction, dimension) and three relational between two vectors (equality, parallelism, orthogonality). The table below collects all six in a single reference, grouped by type, with each property's defining condition and one key fact to remember. The algebraic axioms in the previous section are properties of the operations rather than of vectors themselves, and are developed in full on the vector space axioms page.

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Vectors · summary

Properties at a glance

Three properties belong to a vector on its own; three exist only between a pair. The split matters because the zero vector behaves differently on either side of it.

06properties

Unconditional2

i

Magnitude§ 1

\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}

A single non-negative number: the length of the arrow, independent of where it is drawn.

Unconditional

Requires: intrinsic — belongs to one vector

A single non-negative number: the length of the arrow, independent of where it is drawn. It is zero exactly for the zero vector, which is the property positive definiteness of the dot product records.

iii

Dimension§ 3

\mathbf{v} \in \mathbb{R}^n \;\Rightarrow\; \dim = n

The component count, decided by the space rather than the vector.

Unconditional

Requires: intrinsic — fixed by the space

The component count, decided by the space rather than the vector. No operation on vectors changes it — addition and scaling stay inside $\mathbb{R}^n$ , which is what makes it a vector space rather than merely a set.

Conditional4

ii

Direction§ 2

\hat{\mathbf{v}} = \dfrac{\mathbf{v}}{\|\mathbf{v}\|}

The unit vector pointing the same way.

Conditional

Holds when: $\mathbf{v} \neq \mathbf{0}$
Fails when: $\mathbf{v} = \mathbf{0}$ — division by zero

The unit vector pointing the same way. The zero vector has no direction at all, and this is not a convention that could have gone otherwise: the definition divides by a magnitude of zero. Every later statement about $\mathbf{0}$ traces back here.

iv

Equality§ 4

\mathbf{a} = \mathbf{b} \iff a_i = b_i

for every

i

All-or-nothing: one differing component breaks it.

Conditional

Holds when: both in the same $\mathbb{R}^n$

All-or-nothing: one differing component breaks it. Vectors in different spaces are not unequal so much as incomparable — the question does not arise.

v

Parallelism§ 5

\mathbf{a} \parallel \mathbf{b} \iff \mathbf{a} = c\mathbf{b}

for some scalar

c

A relation between two vectors, not a property of either.

Conditional

Holds when: either vector may be $\mathbf{0}$
Common error: assuming parallel means same direction — $c < 0$ gives opposite directions

A relation between two vectors, not a property of either. The zero vector is parallel to everything by convention, which is consistent because $\mathbf{0} = 0 \cdot \mathbf{b}$ holds for every $\mathbf{b}$ . In $\mathbb{R}^3$ the cross product detects this: parallel vectors have $\mathbf{a} \times \mathbf{b} = \mathbf{0}$ .

vi

Orthogonality§ 6

\mathbf{a} \perp \mathbf{b} \iff \mathbf{a} \cdot \mathbf{b} = 0

Also relational.

Conditional

Holds when: either vector may be $\mathbf{0}$

Also relational. The zero vector is orthogonal to everything, since $\mathbf{0} \cdot \mathbf{b} = 0$ always — so it is simultaneously parallel and orthogonal to every vector, the only one for which both hold. See orthogonality for what this makes possible.

The zero vector case0 · b = 0 for every b → 0 ⊥ b
0 = 0 · b for every b → 0 ∥ b

Unconditional

Conditional

Magnitude§ 1Unconditional

\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}

Requires: intrinsic — belongs to one vector

A single non-negative number: the length of the arrow, independent of where it is drawn. It is zero exactly for the zero vector, which is the property positive definiteness of the dot product records.

Read the full section

Direction§ 2Conditional

\hat{\mathbf{v}} = \dfrac{\mathbf{v}}{\|\mathbf{v}\|}

Holds when: $\mathbf{v} \neq \mathbf{0}$
Fails when: $\mathbf{v} = \mathbf{0}$ — division by zero

The unit vector pointing the same way. The zero vector has no direction at all, and this is not a convention that could have gone otherwise: the definition divides by a magnitude of zero. Every later statement about

\mathbf{0}

traces back here.

Read the full section

Dimension§ 3Unconditional

\mathbf{v} \in \mathbb{R}^n \;\Rightarrow\; \dim = n

Requires: intrinsic — fixed by the space

The component count, decided by the space rather than the vector. No operation on vectors changes it — addition and scaling stay inside

\mathbb{R}^n

, which is what makes it a vector space rather than merely a set.

Read the full section

Equality§ 4Conditional

\mathbf{a} = \mathbf{b} \iff a_i = b_i

for every

i

Holds when: both in the same $\mathbb{R}^n$

All-or-nothing: one differing component breaks it. Vectors in different spaces are not unequal so much as incomparable — the question does not arise.

Read the full section

Parallelism§ 5Conditional

\mathbf{a} \parallel \mathbf{b} \iff \mathbf{a} = c\mathbf{b}

for some scalar

c

Holds when: either vector may be $\mathbf{0}$
Common error: assuming parallel means same direction — $c < 0$ gives opposite directions

A relation between two vectors, not a property of either. The zero vector is parallel to everything by convention, which is consistent because

\mathbf{0} = 0 \cdot \mathbf{b}

holds for every

\mathbf{b}

. In

\mathbb{R}^3

the cross product detects this: parallel vectors have

\mathbf{a} \times \mathbf{b} = \mathbf{0}

.

Read the full section

Orthogonality§ 6Conditional

\mathbf{a} \perp \mathbf{b} \iff \mathbf{a} \cdot \mathbf{b} = 0

Holds when: either vector may be $\mathbf{0}$

Also relational. The zero vector is orthogonal to everything, since

\mathbf{0} \cdot \mathbf{b} = 0

always — so it is simultaneously parallel and orthogonal to every vector, the only one for which both hold. See orthogonality for what this makes possible.

The zero vector case0 · b = 0 for every b → 0 ⊥ b
0 = 0 · b for every b → 0 ∥ b

Read the full section

Notice how often the zero vector is the exception. It has no direction, is parallel to everything, and is orthogonal to everything — three conventions that look arbitrary until you see they all follow from

\mathbf{0}

having magnitude zero and no defined unit vector.

Properties at a glance·/linear-algebra/vectors/propertiesLearn Math Class

Common Pitfalls

Four mistakes account for most wrong answers about vector properties. 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.

Skipping the quadrant adjustment

The direction angle of a vector in

\mathbb{R}^2

is read straight from

\arctan\left(\frac{v_2}{v_1}\right)

, and the signs of the components are never looked at. The arctangent only returns angles between

-90^\circ

and

90^\circ

, so every vector with a negative first component gets an angle on the wrong side.

Wrong working:

\mathbf{v} = (-1, -1), \qquad \alpha = \arctan\left(\frac{-1}{-1}\right) = \arctan(1) = 45^\circ

Right working: both components are negative, so the vector lies in the third quadrant, and

180^\circ

is added to the arctangent.

\alpha = 45^\circ + 180^\circ = 225^\circ

Why it happens: the ratio

\frac{-1}{-1}

equals

1

, the same as

\frac{1}{1}

, so the signs cancel before the arctangent ever sees them.

How to check: the cosine and sine of the angle must have the signs of

v_1

and

v_2

. For

45^\circ

both are positive, which does not match

(-1, -1)

. For

225^\circ

both equal

-\frac{\sqrt{2}}{2}

, which does.

Comparing only the lengths

Two vectors are declared equal because their magnitudes match, or because they contain the same numbers in a different order.

Wrong working:

\|(3, 4)\| = \sqrt{9 + 16} = 5, \quad \|(4, 3)\| = \sqrt{16 + 9} = 5, \quad \text{so } (3, 4) = (4, 3)

Right working: compare the vectors component by component. The first components are

3

and

4

, and

3 \neq 4

, so

(3, 4) \neq (4, 3)

. One differing component is enough.

Why it happens: the magnitude is the single number most often computed for a vector, and it is tempting to let it stand for the whole vector. But a length is shared by many vectors: every vector whose tip lies on the circle of radius

5

has length

5

.

How to check: subtract the two vectors. Equal vectors have difference

\mathbf{0}

, and here

(3, 4) - (4, 3) = (-1, 1) \neq \mathbf{0}

.

Reading not orthogonal as parallel

A nonzero dot product is taken to mean that the two vectors are parallel, as if every pair had to be one or the other.

Wrong working:

(1, 2) \cdot (3, 1) = 3 + 2 = 5 \neq 0, \quad \text{so } (1, 2) \parallel (3, 1)

Right working: parallelism has its own test.

(1, 2) = c(3, 1)

would need

c = \frac{1}{3}

from the first component and

c = 2

from the second, so no such scalar exists. The pair is neither parallel nor orthogonal; the angle between them satisfies

\cos\theta = \frac{5}{\sqrt{5}\,\sqrt{10}} = \frac{1}{\sqrt{2}}, \qquad \theta = 45^\circ

Why it happens: parallelism and orthogonality are introduced together as opposite extremes, and the wide range of angles between them is easy to forget.

How to check: run both tests separately. Parallel vectors are scalar multiples of each other; orthogonal vectors have dot product

0

. A pair that fails both is neither.

Scaling only the first vector

A scalar multiplies a sum of vectors, but it is applied to the first vector only, and the rest are copied unchanged.

Wrong working:

2\big((1, 0) + (0, 3)\big) = (2, 0) + (0, 3) = (2, 3)

Right working: by the distributive law

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

, the scalar reaches every term.

2\big((1, 0) + (0, 3)\big) = (2, 0) + (0, 6) = (2, 6)

Why it happens: the scalar is written right next to the first vector, and the bracket that carries it to the others is easy to lose.

How to check: take the other route and add first.

(1, 0) + (0, 3) = (1, 3)

, and

2(1, 3) = (2, 6)

, which differs from

(2, 3)

.

Vector Properties FAQ

Is the zero vector parallel or orthogonal to other vectors?

+

Both, by convention. Neither is forced by geometry, since

\mathbf{0}

has no direction to align or oppose, but each has a computation behind it:

\mathbf{0} = 0\mathbf{v}

satisfies the definition of parallel, and

\mathbf{0} \cdot \mathbf{v} = 0

satisfies orthogonal. The definitions were written to admit it so that theorems need no exclusion clause.Read more →

Does $\mathbf{a} \parallel \mathbf{b}$ include vectors pointing opposite ways?

+

Yes. The definition allows any scalar multiple, negative ones included, so a vector and its reverse are parallel. The symbol records no distinction between the two cases, which is why the opposite-pointing case needs the separate word anti-parallel. Some texts do reserve

\parallel

for the positive case, so check the convention in use.Read more →

When does the ratio test for parallel vectors break down?

+

Whenever a component is zero, because the test divides by it. Comparing

\frac{a_1}{b_1} = \frac{a_2}{b_2}

works only when no denominator vanishes, yet

(0, 3)

and

(0, 6)

are plainly parallel while the first ratio cannot even be written. Fall back on the definition and look for a scalar

c

with

\mathbf{a} = c\mathbf{b}

.Read more →

Is parallelism transitive?

+

Only through a nonzero vector. If

\mathbf{a} = c\mathbf{b}

and

\mathbf{b} = d\mathbf{w}

, then

\mathbf{a} = (cd)\mathbf{w}

, so parallelism passes along the chain. The zero vector breaks it: by convention

\mathbf{0}

is parallel to everything, so

(1, 0)

and

(0, 1)

are each parallel to

\mathbf{0}

, yet they meet at a right angle. The chain holds whenever the middle vector is nonzero.Read more →

Why does arctan give the wrong angle for $(-1, -1)$ ?

+

Because the ratio

\frac{-1}{-1} = 1

has already lost both signs, and the arctangent only returns angles between

-90^\circ

and

90^\circ

. It answers

45^\circ

, a first-quadrant direction. Both components are negative, so the vector lies in the third quadrant and

180^\circ

is added, giving

225^\circ

. Every vector with a negative first component needs that adjustment; the check is that

\cos\alpha

and

\sin\alpha

carry the signs of

v_1

and

v_2

.Read more →

Can two vectors be neither parallel nor orthogonal?

+

Yes, and most pairs are: parallel and orthogonal are the two extremes, with every other angle in between. A nonzero dot product rules out a right angle but does not make the vectors parallel. For

(1, 2)

and

(3, 1)

the dot product is

5

, yet no scalar

c

gives

(1, 2) = c(3, 1)

, since the components would need

c = \frac{1}{3}

and

c = 2

; the vectors meet at

45^\circ

.Read more →

Does moving a vector to another place change it?

+

No, a free vector is fixed by its length and direction, not by where it is drawn. An arrow at one corner of a diagram and an identical arrow elsewhere are the same vector, as long as length and orientation agree; only rotating or rescaling changes it. That freedom is what lets the tip-to-tail and parallelogram constructions slide arrows into position without altering the result of an addition.Read more →

Properties of a Vector

What Vectors Have and How They Relate

Magnitude

Direction

Dimension

Vector Property Notation

Vector Property Notation

Equality

Parallelism

Orthogonality

Algebraic Properties

Summary: Properties at a Glance

Properties at a glance

Common Pitfalls

Vector Properties FAQ

Is the zero vector parallel or orthogonal to other vectors?

Does $\mathbf{a} \parallel \mathbf{b}$ include vectors pointing opposite ways?

When does the ratio test for parallel vectors break down?

Is parallelism transitive?

Why does arctan give the wrong angle for $(-1, -1)$ ?

Can two vectors be neither parallel nor orthogonal?

Does moving a vector to another place change it?

Properties of a Vector

What Vectors Have and How They Relate

Magnitude

Direction

Dimension

Vector Property Notation

Vector Property Notation

Equality

Parallelism

Orthogonality

Algebraic Properties

Summary: Properties at a Glance

Properties at a glance

Common Pitfalls

Vector Properties FAQ

Is the zero vector parallel or orthogonal to other vectors?

Does a∥b\mathbf{a} \parallel \mathbf{b}a∥b include vectors pointing opposite ways?

When does the ratio test for parallel vectors break down?

Is parallelism transitive?

Why does arctan give the wrong angle for (−1,−1)(-1, -1)(−1,−1)?

Can two vectors be neither parallel nor orthogonal?

Does moving a vector to another place change it?

Does $\mathbf{a} \parallel \mathbf{b}$ include vectors pointing opposite ways?

Why does arctan give the wrong angle for $(-1, -1)$ ?