Vector Basics

A vector is a mathematical object that has both magnitude (size) and direction. Unlike a scalar — which is just a single number like temperature or mass — a vector carries information about which way something points as well as how large it is. Vectors appear everywhere in science and engineering: the velocity of a moving car, the force of gravity on a book, and the displacement between two cities are all vector quantities. In two dimensions, a vector can be thought of as an arrow drawn in the coordinate plane, starting at some point and ending at another. In three dimensions, vectors extend naturally into space. Mathematically, we write a two-dimensional vector as an ordered pair $\mathbf{v} = \langle a, b \rangle$ or as a column matrix, where $a$ and $b$ are called the components of the vector along the $x$ - and $y$ -axes respectively. Vectors are the foundation of linear algebra and are essential for understanding topics ranging from matrix transformations to physics simulations. Mastering vector basics — what they are, how they are classified, and how to represent them — opens the door to a vast and powerful branch of mathematics.

Key takeaways

Representing a Vector
Magnitude (Length) of a Vector
Types of Vectors
Standard Basis Vectors
Direction Angle

Representing a Vector

A vector in $\mathbb{R}^2$ is written as $\mathbf{v} = \langle a, b \rangle$ , where $a$ is the horizontal component and $b$ is the vertical component. Geometrically, the vector is drawn as an arrow from an initial point to a terminal point. If the initial point is $P(x_1, y_1)$ and the terminal point is $Q(x_2, y_2)$ , then the component form is $\mathbf{v} = \langle x_2 - x_1,\; y_2 - y_1 \rangle$ .

Magnitude (Length) of a Vector

The magnitude of a vector $\mathbf{v} = \langle a, b \rangle$ is its length, denoted $|\mathbf{v}|$ or $\|\mathbf{v}\|$ . By the Pythagorean theorem, $\|\mathbf{v}\| = \sqrt{a^2 + b^2}$ . In three dimensions, $\|\mathbf{v}\| = \sqrt{a^2 + b^2 + c^2}$ . Magnitude is always a non-negative scalar.

Types of Vectors

Several special types of vectors are important. A zero vector $\mathbf{0} = \langle 0, 0 \rangle$ has magnitude zero and no defined direction. A unit vector has magnitude exactly 1; any vector $\mathbf{v}$ can be turned into a unit vector by dividing by its magnitude: $\hat{v} = \mathbf{v}/\|\mathbf{v}\|$ . Equal vectors have the same magnitude and direction regardless of position. Opposite vectors have the same magnitude but exactly opposite directions, e.g., $\mathbf{v}$ and $-\mathbf{v}$ . Parallel vectors point in the same or exactly opposite directions and are scalar multiples of each other: $\mathbf{u} = k\mathbf{v}$ for some scalar $k \neq 0$ .

Standard Basis Vectors

In $\mathbb{R}^2$ , the standard basis vectors are $\mathbf{i} = \langle 1, 0 \rangle$ (pointing in the positive $x$ -direction) and $\mathbf{j} = \langle 0, 1 \rangle$ (pointing in the positive $y$ -direction). Any 2D vector can be written as a linear combination: $\langle a, b \rangle = a\,\mathbf{i} + b\,\mathbf{j}$ . In $\mathbb{R}^3$ , a third basis vector $\mathbf{k} = \langle 0, 0, 1 \rangle$ is added, so $\langle a, b, c \rangle = a\,\mathbf{i} + b\,\mathbf{j} + c\,\mathbf{k}$ .

Direction Angle

The direction angle $\theta$ of a vector $\mathbf{v} = \langle a, b \rangle$ is the angle it makes with the positive $x$ -axis, measured counterclockwise. It satisfies $\tan\theta = b/a$ , so $\theta = \arctan(b/a)$ (with quadrant adjustment). The vector can also be expressed as $\mathbf{v} = \|\mathbf{v}\|\langle \cos\theta,\, \sin\theta \rangle$ .

The Key Equation

\|\mathbf{v}\| = \sqrt{a^2 + b^2}

\|\mathbf{v}\| — Magnitude (same unit as components): The length of the vector; always a non-negative real number.
a — Horizontal component (varies): The component of the vector along the x-axis.
b — Vertical component (varies): The component of the vector along the y-axis.

Worked Example

Problem. Let $P = (1, 3)$ and $Q = (5, 6)$ . Find the vector $\overrightarrow{PQ}$ , its magnitude, and the corresponding unit vector.

\overrightarrow{PQ} = \langle 4,3 \rangle,\quad \|\overrightarrow{PQ}\| = 5,\quad \hat{u} = \left\langle \tfrac{4}{5}, \tfrac{3}{5} \right\rangle

Answer. The vector is ⟨4, 3⟩, its magnitude is 5, and the unit vector is ⟨4/5, 3/5⟩.

Figure

Diagram

Origins · 1 minHermann Grassmann and how this was discovered

Hermann Grassmann

1809–1877 · German

Grassmann developed the theory of vector spaces and linear algebra in his 1844 work 'Die lineale Ausdehnungslehre' (The Theory of Linear Extension), independently creating much of what we now call vector algebra. His ideas were far ahead of their time and are the foundation of modern linear algebra and vector analysis. Grassmann's mathematical masterpiece was so abstract and philosophically written that almost no mathematician of his era could understand it. He eventually gave up on mathematics and became a distinguished Sanskrit scholar instead.

Vector Addition and Scalar Multiplication — Once you know what vectors are, the natural next step is learning how to add them together and scale them — the two fundamental operations of linear algebra.
Dot Product — The dot product uses vector components to compute angles between vectors and measure how much two vectors point in the same direction.
Magnitude and Direction — A deeper exploration of how to compute and interpret the length and direction angle of any vector in 2D and 3D.
Introduction to Matrices — Vectors can be represented as column matrices, connecting vector algebra directly to the broader world of matrix operations.