Vectors — Introduction

In short

A vector is a quantity that has both magnitude and direction — unlike a scalar, which has magnitude alone. Vectors are represented as directed line segments (arrows) and classified into types: zero vector, unit vector, equal vectors, negative vectors, coinitial vectors, coterminous vectors, like and unlike vectors, collinear vectors, free vectors, and localised vectors. Every point in space has a position vector from a chosen origin, and this connects the arrow picture to coordinates.

A train travels from Delhi to Jaipur at 120 km/h. Another train also travels at 120 km/h, but from Delhi to Lucknow. Same speed, completely different journeys. The number "120 km/h" does not tell you where either train is going.

This is the problem that creates vectors. Some quantities in the world — temperature, mass, time, speed — are fully described by a single number. But other quantities — velocity, force, displacement, acceleration — need two pieces of information: how much, and in which direction. A number alone is not enough.

The quantities that need only a number are called scalars. The quantities that need both a number and a direction are called vectors. The word "vector" comes from the Latin vehere, to carry — a vector carries information about where, not just how much.

What a vector looks like

A vector is drawn as an arrow. The arrow has two things: a length (representing the magnitude) and a direction (the way it points). The point where the arrow starts is called its initial point (or tail), and the point where it ends is called its terminal point (or head).

A vector from $A$ to $B$, written $\vec{AB}$. The arrow carries two pieces of information: its length (the magnitude, written $|\vec{AB}|$) and its direction (from $A$ toward $B$). Reverse the arrow and you get a different vector: same magnitude, opposite direction.

Notation. A vector from point A to point B is written \vec{AB}. When you use a single letter for a vector, you write it in bold (\mathbf{a}) in print, or with an arrow on top (\vec{a}) by hand. The magnitude of \vec{a} is written |\vec{a}| or just a (without bold or arrow).

Scalars vs vectors — a summary:

Scalar	Vector
Has magnitude only	Has magnitude and direction
Temperature: 37°C	Displacement: 5 km north
Mass: 60 kg	Velocity: 120 km/h toward Jaipur
Speed: 30 m/s	Force: 10 N downward
Distance: 200 m	Acceleration: 9.8 m/s² downward

Notice that speed is a scalar but velocity is a vector. Distance is a scalar but displacement is a vector. The vector versions carry directional information that the scalar versions discard.

Types of vectors

Vectors are classified into several types based on their magnitude, direction, and how they relate to each other. Each type captures a different geometric idea.

Zero vector

The zero vector, written \vec{0}, has magnitude zero. Its initial and terminal points coincide — the arrow has no length. Its direction is undefined (or, equivalently, considered to be in every direction at once).

You might wonder: why bother with a vector of zero length? For the same reason you bother with the number zero in arithmetic — it is the identity element for addition. When you add any vector \vec{a} to the zero vector, you get \vec{a} back unchanged: \vec{a} + \vec{0} = \vec{a}. It plays a structural role.

Unit vector

A unit vector is any vector whose magnitude is exactly 1. If \vec{a} is a non-zero vector, then

\hat{a} = \frac{\vec{a}}{|\vec{a}|}

is the unit vector in the direction of \vec{a}. The hat symbol \hat{a} (read "a-hat") signals "unit vector." Dividing by the magnitude strips away the length and leaves only the direction.

The three standard unit vectors along the coordinate axes are \hat{i} (along the x-axis), \hat{j} (along the y-axis), and \hat{k} (along the z-axis). Every vector in three-dimensional space can be written as a combination of these three.

The vector $\vec{a}$ and its unit vector $\hat{a}$. Both point in the same direction, but $\hat{a}$ has magnitude $1$. Dividing a vector by its magnitude is like normalising: you keep the direction and throw away the scale. Every non-zero vector has a unique unit vector.

Equal vectors

Two vectors are equal if they have the same magnitude and the same direction. Their positions in space do not matter — only the length and the direction. A vector drawn from (0, 0) to (3, 4) and another drawn from (5, 1) to (8, 5) are equal vectors, because both have the same length (5) and point in the same direction.

This is a key idea: vectors are not anchored to a specific location (unless they are localised vectors, which we come to below). Two arrows in different parts of the page can represent the same vector.

Negative of a vector

The negative of a vector \vec{a}, written -\vec{a}, has the same magnitude as \vec{a} but the opposite direction. If \vec{a} points northeast, then -\vec{a} points southwest, with the same length.

Equivalently, \vec{AB} = -\vec{BA}. Reversing the order of the points reverses the arrow.

A vector and its negative. They cancel when added: $\vec{a} + (-\vec{a}) = \vec{0}$. The negative of a vector reverses its direction but preserves its magnitude.

Coinitial vectors

Two or more vectors are coinitial if they share the same initial point — they all start from the same tail. For instance, \vec{OA} and \vec{OB} are coinitial at O.

Coterminous vectors

Two or more vectors are coterminous if they share the same terminal point — they all end at the same head.

Collinear (parallel) vectors

Two vectors are collinear (also called parallel) if they lie along the same line, or along parallel lines. They may point in the same direction or in opposite directions. Algebraically, \vec{a} and \vec{b} are collinear if \vec{b} = \lambda\vec{a} for some scalar \lambda.

Like and unlike vectors

Among collinear vectors, those pointing in the same direction are called like vectors, and those pointing in opposite directions are called unlike vectors. Like vectors have \lambda > 0 in the relation \vec{b} = \lambda\vec{a}; unlike vectors have \lambda < 0.

The classification of vector pairs. Like vectors point the same way (both rightward). Unlike vectors point opposite ways along the same line. Non-collinear vectors point in different directions entirely — no scalar multiple of one gives the other.

Free vectors and localised vectors

A free vector is a vector defined only by its magnitude and direction — it can be placed anywhere in space without changing what it represents. Most vectors in mathematics are free vectors. When you say "\vec{a} and \vec{b} are equal," you are treating them as free vectors: their starting point does not matter.

A localised vector (also called a bound vector) is tied to a specific point. Force in physics is a localised vector — a force of 10 N applied at the edge of a door produces a different effect than the same force applied at the hinge, even though the magnitude and direction are identical. Where you apply it matters.

In this article and in most of the vector algebra you will study, vectors are free unless stated otherwise.

Position vector

Here is where vectors connect to coordinates.

Pick a fixed point O as the origin. For any point P in space, the vector \vec{OP} — the arrow from the origin to P — is called the position vector of P. It encodes the location of P relative to the origin.

If P has coordinates (x, y, z), then its position vector is

\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}

The components x, y, z are the projections of the arrow onto the three axes.

The position vector $\vec{OP}$ of the point $P(3, 2, 4)$. The red arrow from the origin to $P$ is the vector $3\hat{i} + 2\hat{j} + 4\hat{k}$. The dashed lines show the components: $3$ units along $x$, $2$ units along $y$, $4$ units along $z$. Every point in space has a unique position vector from a chosen origin.

Magnitude from components

The magnitude of \vec{OP} = x\hat{i} + y\hat{j} + z\hat{k} is the distance from the origin to P:

|\vec{OP}| = \sqrt{x^2 + y^2 + z^2}

This is the three-dimensional version of the Pythagorean theorem. For the point P(3, 2, 4): |\vec{OP}| = \sqrt{9 + 4 + 16} = \sqrt{29}.

In two dimensions, the position vector of P(x, y) is x\hat{i} + y\hat{j}, with magnitude \sqrt{x^2 + y^2}.

Displacement vector from position vectors

If A has position vector \vec{OA} and B has position vector \vec{OB}, then the vector from A to B is:

\vec{AB} = \vec{OB} - \vec{OA}

This is the vector equivalent of "subtract the starting point from the ending point." If A = (1, 2, 3) and B = (4, 6, 3), then \vec{AB} = (4-1)\hat{i} + (6-2)\hat{j} + (3-3)\hat{k} = 3\hat{i} + 4\hat{j}.

The magnitude is |\vec{AB}| = \sqrt{9 + 16} = 5 — which is the distance between A and B. Position vectors turn distance calculations into vector subtraction.

Formal definition

Vector

A vector is a quantity characterised by both a magnitude (a non-negative real number) and a direction in space. Two vectors are equal if and only if they have the same magnitude and the same direction.

A vector is represented geometrically as a directed line segment (arrow). Algebraically, in a coordinate system with origin O, the position vector of a point P(x, y, z) is \vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}, and its magnitude is |\vec{OP}| = \sqrt{x^2 + y^2 + z^2}.

Reading the definition. The key word is both. Temperature is 37°C — one number, no direction, a scalar. But "5 km north" is a displacement — it has a size (5 km) and a direction (north). Strip the direction and you are left with a distance, not a displacement. The direction is not optional; it is half the information.

Worked examples

Example 1: Find the unit vector in the direction of $\vec{a} = 2\hat{i} - 3\hat{j} + 6\hat{k}$

Step 1. Compute the magnitude.

|\vec{a}| = \sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7

Why: the magnitude is the 3D Pythagorean distance. The components 2, -3, 6 are chosen to give a clean integer magnitude — 2^2 + 3^2 + 6^2 = 49.

Step 2. Divide each component by the magnitude.

\hat{a} = \frac{\vec{a}}{|\vec{a}|} = \frac{2\hat{i} - 3\hat{j} + 6\hat{k}}{7} = \frac{2}{7}\hat{i} - \frac{3}{7}\hat{j} + \frac{6}{7}\hat{k}

Why: dividing by the magnitude normalises the vector to length 1 while preserving direction.

Step 3. Verify: |\hat{a}| = \sqrt{(2/7)^2 + (-3/7)^2 + (6/7)^2} = \sqrt{4/49 + 9/49 + 36/49} = \sqrt{49/49} = 1.

Why: a unit vector must have magnitude exactly 1. This confirms the division worked correctly.

Step 4. Interpret. The unit vector \hat{a} points in the same direction as \vec{a} but has length 1. It encodes pure direction with no scale.

Result: \hat{a} = \frac{2}{7}\hat{i} - \frac{3}{7}\hat{j} + \frac{6}{7}\hat{k}.

The vector $\vec{a} = 2\hat{i} - 3\hat{j} + 6\hat{k}$ (long arrow, magnitude $7$) and its unit vector $\hat{a}$ (short arrow, magnitude $1$). Both point in the same direction. The unit vector is exactly $1/7$th the length of the original. Any non-zero vector, no matter how large, can be reduced to a unit vector by this division.

The triple (2, -3, 6) was chosen because 2^2 + 3^2 + 6^2 = 49 = 7^2, giving a clean magnitude. In practice, magnitudes are rarely this tidy — but the method is the same regardless.

Example 2: Show that the points $A(1, 2, 3)$, $B(3, 4, 5)$, and $C(5, 6, 7)$ are collinear using vectors

Step 1. Compute \vec{AB}.

\vec{AB} = \vec{OB} - \vec{OA} = (3-1)\hat{i} + (4-2)\hat{j} + (5-3)\hat{k} = 2\hat{i} + 2\hat{j} + 2\hat{k}

Why: the displacement from A to B is found by subtracting position vectors component by component.

Step 2. Compute \vec{AC}.

\vec{AC} = \vec{OC} - \vec{OA} = (5-1)\hat{i} + (6-2)\hat{j} + (7-3)\hat{k} = 4\hat{i} + 4\hat{j} + 4\hat{k}

Why: same method — subtract the coordinates of A from the coordinates of C.

Step 3. Check if \vec{AC} is a scalar multiple of \vec{AB}.

\vec{AC} = 4\hat{i} + 4\hat{j} + 4\hat{k} = 2(2\hat{i} + 2\hat{j} + 2\hat{k}) = 2\vec{AB}

Why: if \vec{AC} = \lambda\vec{AB} for some scalar \lambda, then \vec{AC} and \vec{AB} are collinear — they lie along the same line. Here \lambda = 2.

Step 4. Since \vec{AB} and \vec{AC} are collinear and share the initial point A, the points A, B, C all lie on the same line. Moreover, \lambda = 2 > 0 tells you that C is on the same side of A as B (like vectors), and |\vec{AC}| = 2|\vec{AB}| tells you that C is twice as far from A as B is — meaning B is the midpoint of AC.

Result: A(1,2,3), B(3,4,5), C(5,6,7) are collinear, with B as the midpoint.

The three points $A$, $B$, $C$ projected onto 2D (showing just the $x$ and $y$ coordinates). They lie on the line $y = x + 1$. The vector $\vec{AB} = (2, 2)$ and the vector $\vec{BC} = (2, 2)$ are equal — $B$ is the midpoint, and all three points are collinear. The vector test ($\vec{AC} = 2\vec{AB}$) confirms what the picture shows.

The collinearity test — check if one displacement vector is a scalar multiple of another — is the vector way to decide if three points lie on a line. It generalises to any dimension and avoids the formula-heavy approach of checking slopes.

Common confusions

"A vector has a fixed position." Not by default. Free vectors are defined by magnitude and direction only — they can be placed anywhere in space. Two arrows at different locations can represent the same vector. The only vectors anchored to specific points are localised vectors (like forces applied at specific points) and position vectors (which are anchored to the origin by definition).
"The zero vector has a direction." It does not. The direction of the zero vector is undefined. Some textbooks say it has "any direction" or "every direction," but the honest statement is that direction is not a meaningful concept when the magnitude is zero. This is analogous to 0/0 being undefined — there is no single answer.
"Speed and velocity are the same thing." Speed is a scalar (magnitude only). Velocity is a vector (magnitude and direction). A car moving at 60 km/h in a circle has constant speed but changing velocity, because the direction keeps changing. This distinction is the whole reason vectors exist.
"Two vectors with the same magnitude are equal." No — they must also have the same direction. 3\hat{i} and 3\hat{j} have the same magnitude (3) but point in perpendicular directions. They are not equal.
"Collinear vectors must be equal." Collinear vectors lie along the same line, but they can have different magnitudes and can point in opposite directions (unlike vectors). 2\hat{i} and -5\hat{i} are collinear but neither equal nor opposite.

Going deeper

If you came here to understand what vectors are, how to classify them, and how position vectors connect arrows to coordinates, you have it — you can stop here. The rest of this section is for readers who want to see the abstract framework behind vectors.

Why direction matters: a physics perspective

In physics, forces add as vectors, not as numbers. If two people push a box — one with 5 N eastward and one with 5 N northward — the total force is not 10 N. It is 5\sqrt{2} \approx 7.07 N in the northeast direction. The Pythagorean theorem enters because the two forces are perpendicular. If they pushed in the same direction, the total would be 10 N; if in opposite directions, 0 N. The total depends on the angle between them.

This is why scalars cannot model forces. The number 5 does not tell you which way the force points, and without direction, you cannot compute the result of combining forces. Vectors were invented precisely to solve this problem.

Vectors as equivalence classes

Mathematically, a free vector is not one specific arrow — it is the entire collection of all arrows with the same length and direction. Two arrows are "equivalent" if they have the same length and point the same way, regardless of where they are placed. Each vector is an equivalence class under this relation.

The position vector \vec{OP} is the representative of this class that starts at the origin. This is why position vectors are special: every free vector has exactly one representative that starts at O, and that representative is uniquely determined by the coordinates of its head. This is the bridge between the geometric picture (arrows floating freely) and the algebraic representation (ordered tuples of numbers).

The vector space axioms

The collection of all position vectors in \mathbb{R}^n forms what mathematicians call a vector space. The two operations — addition and scalar multiplication — satisfy a set of axioms (closure, associativity, commutativity, distributivity, existence of zero vector, existence of additive inverse). You will study these axioms in detail in linear algebra. For now, the key point is that vectors are not just useful objects — they are the foundational objects of one of the most important structures in all of mathematics.

Connection to complex numbers

A complex number a + bi looks remarkably like a 2D position vector a\hat{i} + b\hat{j}. This is not a coincidence. The complex plane is a 2D vector space, with complex addition matching vector addition and |z| matching |\vec{v}|. The extra structure of complex multiplication (which vectors in general do not have) is what makes complex numbers special. The article on complex numbers explores this connection.

Where this leads next

Vector Operations — how to add vectors (triangle law, parallelogram law), subtract them, and multiply by a scalar, with proofs of the key properties.
Components and Direction — resolving a vector into components along the axes, direction cosines, and direction ratios.
Coordinate Geometry Basics — the coordinate system that position vectors live in.
Trigonometric Ratios — the sine and cosine that appear when you resolve vectors into components along non-axis directions.
Complex Numbers — Introduction — the 2D number system that shares the structure of 2D vectors, with one powerful extra operation.