Vectors in \(\mathbb{R}^n\)

Learn how to represent, add, and scale ordered lists of numbers that form the foundation of all linear algebra.

Linear Algebra

What Is a Vector?

A vector is an ordered list of numbers. In \(\mathbb{R}^n\), a vector has exactly \(n\) real-number entries called components. We write vectors as columns:

For example, \(\begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}\) is a vector in \(\mathbb{R}^3\) with three components.

Visualizing Vectors in \(\mathbb{R}^2\)

In \(\mathbb{R}^2\), each vector is an arrow from the origin to a point \((x, y)\). The vector \(\mathbf{v} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\) points to the coordinate (3, 2).

Why Study \(\mathbb{R}^n\)?

Physics

Force, velocity, and acceleration are vectors in \(\mathbb{R}^3\).

Data Science

A data point with \(n\) features is a vector in \(\mathbb{R}^n\).

Graphics

3D rendering uses vectors in \(\mathbb{R}^3\) and \(\mathbb{R}^4\).

Equality of Vectors

This means \(\begin{pmatrix} 1 \\ 2 \end{pmatrix} \neq \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}\) because they live in different spaces: the first is in \(\mathbb{R}^2\) and the second in \(\mathbb{R}^3\).

Vector Addition

Geometrically, place the tail of \(\mathbf{v}\) at the tip of \(\mathbf{u}\): the diagonal of the parallelogram is \(\mathbf{u}+\mathbf{v}\).

Scalar Multiplication

Multiplying by \(c > 1\) stretches the vector; by \(0 < c < 1\) shrinks it; by \(c < 0\) reverses direction.

Algebraic Properties

For all \(\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^n\) and scalars \(c, d \in \mathbb{R}\):

1

Commutativity: \(\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\)

2

Associativity: \((\mathbf{u}+\mathbf{v})+\mathbf{w} = \mathbf{u}+(\mathbf{v}+\mathbf{w})\)

3

Additive identity: \(\mathbf{u} + \mathbf{0} = \mathbf{u}\) where \(\mathbf{0} = (0, \ldots, 0)\)

4

Additive inverse: \(\mathbf{u} + (-\mathbf{u}) = \mathbf{0}\)

5

Distributive: \(c(\mathbf{u}+\mathbf{v}) = c\mathbf{u} + c\mathbf{v}\) and \((c+d)\mathbf{u} = c\mathbf{u} + d\mathbf{u}\)

The Zero Vector and Negation

Zero Vector \(\mathbf{0}\)

The vector whose every component is 0. It lives in whichever \(\mathbb{R}^n\) the context requires: \(\mathbf{0} \in \mathbb{R}^3\) means \(\begin{pmatrix}0\\0\\0\end{pmatrix}\).

Negative \(-\mathbf{v}\)

Negate every component: if \(\mathbf{v} = \begin{pmatrix}2\\-5\\1\end{pmatrix}\), then \(-\mathbf{v} = \begin{pmatrix}-2\\5\\-1\end{pmatrix}\). Equivalently, \((-1)\mathbf{v}\).

Vectors vs Scalars vs Points

	Vector	Scalar	Point
Nature	Ordered list of numbers	Single number	Location in space
Notation	\(\mathbf{v}\) (bold) or \(\vec{v}\)	\(c, \lambda, \alpha\)	\(P = (x, y)\)
Has direction?	Yes	No	No
Can be added?	Yes (to another vector)	Yes (to another scalar)	Not directly

A point names a location; a vector describes a displacement. The point (3,2) and the vector \(\begin{pmatrix}3\\2\end{pmatrix}\) look the same numerically but carry different meanings.

How to Compute Vector Operations

1

Check dimensions: both vectors must be in the same \(\mathbb{R}^n\) before adding or subtracting.

2

Line up components: write the vectors as columns, aligning corresponding entries.

3

Operate entry-by-entry: add, subtract, or multiply each component by the scalar.

4

Write the result as a new vector in the same \(\mathbb{R}^n\).

Example 1: Addition in \(\mathbb{R}^3\)

Compute \(\mathbf{u} + \mathbf{v}\) where \(\mathbf{u} = \begin{pmatrix}2\\-1\\5\end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix}-3\\4\\1\end{pmatrix}\).

Example 2: Scalar Multiplication

Let \(\mathbf{w} = \begin{pmatrix}4\\0\\-2\\7\end{pmatrix}\). Compute \(-3\,\mathbf{w}\).

Notice the result is still in \(\mathbb{R}^4\). The zero component stays zero regardless of the scalar.

Example 3: Linear Combination

Compute \(2\mathbf{u} - 3\mathbf{v}\) where \(\mathbf{u} = \begin{pmatrix}1\\-2\\3\end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix}0\\4\\-1\end{pmatrix}\).

Common Mistakes

✕ Adding Different Dimensions

You cannot add a vector in \(\mathbb{R}^2\) to one in \(\mathbb{R}^3\). Always check that both vectors have the same number of components.

✕ Multiplying Two Vectors

There is no "multiplication" of two vectors by default. The dot product and cross product are separate operations with different rules.

✕ Forgetting Sign Flips

When computing \(-\mathbf{v}\) or subtracting, students often flip only the first component. Every component must be negated.

Operations Summary

Operation	Formula	Result Space
Addition	\(\mathbf{u}+\mathbf{v}\): add entry-by-entry	\(\mathbb{R}^n\)
Subtraction	\(\mathbf{u}-\mathbf{v} = \mathbf{u}+(-\mathbf{v})\)	\(\mathbb{R}^n\)
Scalar mult.	\(c\,\mathbf{v}\): multiply each entry by \(c\)	\(\mathbb{R}^n\)
Linear combo.	\(c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k\)	\(\mathbb{R}^n\)
Zero vector	\(\mathbf{0} = (0,\ldots,0)\)	\(\mathbb{R}^n\)

Question 1 of 5

What is \(\begin{pmatrix}2\\-1\\4\end{pmatrix} + \begin{pmatrix}5\\3\\-2\end{pmatrix}\)?

Add component-wise: \(2+5=7\), \(-1+3=2\), \(4+(-2)=2\). The result is \(\begin{pmatrix}7\\2\\2\end{pmatrix}\).

Question 2 of 5

Compute \(-2\begin{pmatrix}3\\0\\-4\end{pmatrix}\).

Multiply every component by \(-2\): \(-2 \cdot 3 = -6\), \(-2 \cdot 0 = 0\), \(-2 \cdot (-4) = 8\). The result is \(\begin{pmatrix}-6\\0\\8\end{pmatrix}\).

Question 3 of 5

Which property does vector addition in \(\mathbb{R}^n\) not satisfy?

Vector division is not defined. There is no component-wise division operation that produces a meaningful vector result. The other three are standard axioms of a vector space.

Question 4 of 5

Find \(3\mathbf{u} - \mathbf{v}\) where \(\mathbf{u} = \begin{pmatrix}1\\2\end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix}-1\\5\end{pmatrix}\).

First compute \(3\mathbf{u} = \begin{pmatrix}3\\6\end{pmatrix}\). Then subtract: \(\begin{pmatrix}3-(-1)\\6-5\end{pmatrix} = \begin{pmatrix}4\\1\end{pmatrix}\).

Question 5 of 5

A vector \(\mathbf{v}\) has 5 components. Which space does \(\mathbf{v}\) belong to?

The superscript in \(\mathbb{R}^n\) tells you the number of components. A vector with 5 components lives in \(\mathbb{R}^5\).