Chapter 3. Geometry of Euclidean Space

Euclidean space is the geometric model behind the coordinate spaces $\mathbb{R}^1$ , $\mathbb{R}^2$ , $\mathbb{R}^3$ , and more generally $\mathbb{R}^n$ . It supplies the usual notions of length, distance, angle, line, plane, and orthogonality.

Linear algebra gives these geometric ideas an algebraic form. A point becomes a list of coordinates. A displacement becomes a vector. Length is computed from a dot product. Distance is computed from the difference of two vectors. Orthogonality is expressed by a zero dot product. These definitions allow geometry to be used in any finite dimension, not only in the plane or in ordinary three-dimensional space. The dot product of two coordinate vectors is the sum of products of corresponding coordinates, and it is the standard tool used to define length and angle in Euclidean coordinate space.

3.1 Points and Coordinates

The set $\mathbb{R}^n$ consists of all ordered $n$ -tuples of real numbers:

(x_1,x_2,\ldots,x_n).

Each entry is a coordinate. The number $n$ is the dimension of the coordinate space.

For example, $\mathbb{R}^2$ is the plane. A point in $\mathbb{R}^2$ has the form

(x,y).

The space $\mathbb{R}^3$ is ordinary three-dimensional coordinate space. A point in $\mathbb{R}^3$ has the form

(x,y,z).

The space $\mathbb{R}^n$ generalizes this idea. A point in $\mathbb{R}^n$ has $n$ real coordinates.

3.2 Points and Vectors

A point describes a location. A vector describes a displacement.

In coordinates, both may be written as lists of numbers. The distinction is conceptual. The point

p = \begin{bmatrix} 2 \\ 3 \end{bmatrix}

marks a location in the plane. The vector

v = \begin{bmatrix} 2 \\ 3 \end{bmatrix}

may be interpreted as the displacement that moves $2$ units in the first coordinate direction and $3$ units in the second coordinate direction.

The same coordinate list can therefore represent a point or a vector, depending on context. Linear algebra usually treats coordinate lists as vectors. Geometry often shifts between both views.

3.3 Displacement Between Points

Given two points $p,q \in \mathbb{R}^n$ , the displacement from $p$ to $q$ is

q-p.

p = \begin{bmatrix} p_1 \\ p_2 \\ \vdots \\ p_n \end{bmatrix}, \qquad q = \begin{bmatrix} q_1 \\ q_2 \\ \vdots \\ q_n \end{bmatrix},

then

q-p = \begin{bmatrix} q_1-p_1 \\ q_2-p_2 \\ \vdots \\ q_n-p_n \end{bmatrix}.

For example, if

p = \begin{bmatrix} 1 \\ 4 \end{bmatrix}, \qquad q = \begin{bmatrix} 5 \\ 2 \end{bmatrix},

then

q-p = \begin{bmatrix} 4 \\ -2 \end{bmatrix}.

This vector says that one moves $4$ units in the first coordinate direction and $-2$ units in the second coordinate direction.

3.4 The Dot Product

The dot product of two vectors in $\mathbb{R}^n$ is defined by

u \cdot v = u_1v_1 + u_2v_2 + \cdots + u_nv_n.

u = \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{bmatrix}, \qquad v = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix},

then

u \cdot v = \sum_{i=1}^{n} u_i v_i.

The dot product takes two vectors and returns a scalar. It is therefore also called the scalar product. In Euclidean geometry, it connects coordinates with length and angle.

For example,

\begin{bmatrix} 1 \\ 3 \\ -5 \end{bmatrix} \cdot \begin{bmatrix} 4 \\ -2 \\ -1 \end{bmatrix} = 1\cdot 4 + 3\cdot(-2) + (-5)\cdot(-1) = 3.

3.5 Length of a Vector

The length, or Euclidean norm, of a vector $v\in\mathbb{R}^n$ is defined by

\|v\| = \sqrt{v\cdot v}.

v = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix},

then

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

This is the higher-dimensional form of the Pythagorean theorem. The Euclidean norm is interpreted as the length of the vector.

For example,

v = \begin{bmatrix} 3 \\ 4 \end{bmatrix}

has length

\|v\| = \sqrt{3^2+4^2} = 5.

3.6 Distance Between Points

The distance between two points $p,q \in \mathbb{R}^n$ is the length of the displacement vector $q-p$ :

d(p,q) = \|q-p\|.

Thus

d(p,q) = \sqrt{(q_1-p_1)^2+(q_2-p_2)^2+\cdots+(q_n-p_n)^2}.

For example, if

p = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \qquad q = \begin{bmatrix} 4 \\ 6 \end{bmatrix},

then

q-p = \begin{bmatrix} 3 \\ 4 \end{bmatrix},

and

d(p,q)=5.

Distance is therefore a derived notion. It comes from subtraction of points followed by the Euclidean norm.

3.7 Unit Vectors

A unit vector is a vector of length $1$ .

If $v\ne 0$ , then

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

is a unit vector in the same direction as $v$ .

For example, if

v = \begin{bmatrix} 3 \\ 4 \end{bmatrix},

then $\|v\|=5$ , so

\frac{v}{\|v\|} = \begin{bmatrix} 3/5 \\ 4/5 \end{bmatrix}.

This vector has length $1$ . It preserves direction and removes scale.

3.8 Angles

The angle $\theta$ between two nonzero vectors $u$ and $v$ is defined by

\cos\theta = \frac{u\cdot v}{\|u\|\|v\|}.

Equivalently,

u\cdot v = \|u\|\|v\|\cos\theta.

This formula shows the geometric meaning of the dot product. It measures how much two vectors point in the same direction.

If $u\cdot v>0$ , the angle is acute. If $u\cdot v<0$ , the angle is obtuse. If $u\cdot v=0$ , the angle is right.

3.9 Orthogonality

Two vectors $u$ and $v$ are orthogonal if

u\cdot v = 0.

Geometrically, nonzero orthogonal vectors meet at a right angle. The zero vector is orthogonal to every vector because

0\cdot v = 0.

For example,

\begin{bmatrix} 1 \\ 2 \end{bmatrix} \cdot \begin{bmatrix} 2 \\ -1 \end{bmatrix} = 1\cdot 2 + 2\cdot(-1) = 0.

Therefore the two vectors are orthogonal.

Orthogonality is one of the central ideas of Euclidean geometry and linear algebra. It appears in projections, least squares, QR decomposition, Fourier series, and numerical methods.

3.10 Lines in Euclidean Space

A line through a point $p$ in direction $v\ne 0$ is the set

L = \{p + tv : t\in\mathbb{R}\}.

The parameter $t$ moves along the line. When $t=0$ , the point is $p$ . When $t=1$ , the point is $p+v$ . Negative values of $t$ move in the opposite direction.

For example, the line through

p = \begin{bmatrix} 1 \\ 2 \end{bmatrix}

with direction

v = \begin{bmatrix} 3 \\ -1 \end{bmatrix}

\begin{bmatrix} 1 \\ 2 \end{bmatrix} + t \begin{bmatrix} 3 \\ -1 \end{bmatrix} = \begin{bmatrix} 1+3t \\ 2-t \end{bmatrix}.

A line is therefore an affine set: it is formed by translating a one-dimensional subspace.

3.11 Planes

A plane in $\mathbb{R}^3$ can be described by a point and two independent direction vectors:

P = \{p + su + tv : s,t\in\mathbb{R}\}.

Here $u$ and $v$ must not be scalar multiples of each other. They give two independent directions inside the plane.

A plane may also be described by a normal vector. If $n\ne 0$ is perpendicular to the plane and $p$ is a point on the plane, then

n\cdot(x-p)=0.

This equation says that the displacement from $p$ to any point $x$ on the plane is orthogonal to $n$ .

n = \begin{bmatrix} a \\ b \\ c \end{bmatrix}, \qquad p = \begin{bmatrix} p_1 \\ p_2 \\ p_3 \end{bmatrix},

then the plane equation becomes

a(x_1-p_1)+b(x_2-p_2)+c(x_3-p_3)=0.

Equivalently,

ax_1+bx_2+cx_3=d

for some scalar $d$ .

3.12 Hyperplanes

In $\mathbb{R}^n$ , a hyperplane is the higher-dimensional analogue of a line in $\mathbb{R}^2$ or a plane in $\mathbb{R}^3$ .

A hyperplane can be written as

H = \{x\in\mathbb{R}^n : a\cdot x = b\},

where $a\ne 0$ . The vector $a$ is normal to the hyperplane.

If $b=0$ , the hyperplane passes through the origin and is a subspace. If $b\ne 0$ , the hyperplane is an affine set.

Hyperplanes occur throughout linear algebra. Each linear equation

a_1x_1+a_2x_2+\cdots+a_nx_n=b

defines a hyperplane in $\mathbb{R}^n$ . A system of linear equations describes the intersection of several hyperplanes.

3.13 Projections

The projection of a vector $v$ onto a nonzero vector $u$ is the component of $v$ in the direction of $u$ .

It is given by

\operatorname{proj}_u(v) = \frac{v\cdot u}{u\cdot u}u.

The scalar

\frac{v\cdot u}{u\cdot u}

measures how much of $u$ is needed to form the projection.

For example, let

v = \begin{bmatrix} 3 \\ 4 \end{bmatrix}, \qquad u = \begin{bmatrix} 1 \\ 0 \end{bmatrix}.

Then

v\cdot u = 3, \qquad u\cdot u = 1,

\operatorname{proj}_u(v) = 3 \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \end{bmatrix}.

Projection separates a vector into a part along a direction and a part orthogonal to that direction.

3.14 Orthogonal Decomposition

Let $u\ne 0$ . Every vector $v$ can be decomposed as

v = p + r,

where $p$ is parallel to $u$ and $r$ is orthogonal to $u$ .

The parallel part is

p = \operatorname{proj}_u(v).

The orthogonal part is

r = v - \operatorname{proj}_u(v).

Then

r\cdot u = 0.

This decomposition is fundamental. It is the geometric basis of least squares approximation, orthogonal projection onto subspaces, and many numerical algorithms.

3.15 Spheres and Balls

A sphere in $\mathbb{R}^n$ with center $c$ and radius $r>0$ is the set

S = \{x\in\mathbb{R}^n : \|x-c\| = r\}.

A ball is the set of points whose distance from the center is less than or equal to the radius:

B = \{x\in\mathbb{R}^n : \|x-c\| \le r\}.

In $\mathbb{R}^2$ , a sphere is a circle and a ball is a disk. In $\mathbb{R}^3$ , a sphere is the usual surface of a sphere and a ball is the solid region inside it.

The formulas remain valid in all dimensions.

3.16 Geometry and Linear Equations

A linear equation in $n$ variables,

a_1x_1+a_2x_2+\cdots+a_nx_n=b,

can be written as

a\cdot x=b,

where

a = \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{bmatrix}.

Thus each linear equation defines a hyperplane with normal vector $a$ .

A system of equations

Ax=b

describes the intersection of hyperplanes. Depending on the arrangement, the intersection may be empty, a point, a line, a plane, or a higher-dimensional affine set.

This geometric view explains why systems of linear equations may have no solution, one solution, or infinitely many solutions.

3.17 Euclidean Geometry in Matrix Form

Matrices act on Euclidean space by sending vectors to vectors.

If $A$ is an $m\times n$ matrix, then the rule

x \mapsto Ax

maps $\mathbb{R}^n$ into $\mathbb{R}^m$ .

Some matrices preserve Euclidean geometry. For example, an orthogonal matrix $Q$ satisfies

Q^TQ = I.

Such a matrix preserves dot products:

(Qu)\cdot(Qv)=u\cdot v.

Therefore it preserves lengths and angles. Rotations and reflections are examples of orthogonal transformations.

Other matrices distort geometry. They may stretch, shrink, shear, or collapse dimensions. Much of linear algebra studies these changes by examining rank, determinant, eigenvalues, singular values, and invariant subspaces.

3.18 Summary

Euclidean space combines coordinates with geometry. Points describe locations. Vectors describe displacements. The dot product gives algebraic definitions of length, distance, angle, and orthogonality. Lines, planes, and hyperplanes are described using vectors and linear equations.

The main formulas are:

Concept	Formula
Dot product	$u\cdot v=\sum_{i=1}^{n}u_iv_i$
Norm	$\\|v\\|=\sqrt{v\cdot v}$
Distance	$d(p,q)=\\|q-p\\|$
Angle	$\cos\theta=\frac{u\cdot v}{\\|u\\|\\|v\\|}$
Orthogonality	$u\cdot v=0$
Line	$p+tv$
Hyperplane	$a\cdot x=b$
Projection	$\operatorname{proj}_u(v)=\frac{v\cdot u}{u\cdot u}u$

Euclidean geometry is therefore not separate from linear algebra. It is one of its basic interpretations.