Theory

Vectors in the plane and in space

What a geometric vector is, how to compute with components, and how the dot product, the cross product and the scalar triple product measure angles, areas and volumes and detect orthogonality, parallelism and coplanarity.

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Geometric vectors and operations
Components in a Cartesian coordinate system

Geometric vectors and operations

1. Geometric vector

A directed segment $A B$ is a segment with a first endpoint $A$ (the tail) and a second endpoint $B$ (the head). Two directed segments are called equipollent if they have the same length, the same direction (that is, they lie on parallel lines) and the same orientation. A geometric vector is the class of all directed segments equipollent to a given one; the vector determined by $A B$ is denoted by $A B$ , a generic vector by $v$ .

A vector therefore has three attributes: length, direction and orientation. The starting point is free, since given a point $P$ and a vector $v$ there exists exactly one point $Q$ such that $P Q = v$ ; one then says that $v$ is applied at $P$ . The zero vector $0$ is represented by the segments whose tail and head coincide, $0 = AA$ ; it has length zero and undefined direction.

2. Sum of vectors

Let $v$ and $w$ be two vectors. Apply $v$ at a point $A$ , obtaining $v = A B$ , and apply $w$ at the head $B$ , obtaining $w = B C$ . The sum is

v + w = A C .

This is the head-to-tail rule; when $v$ and $w$ are applied at the same point and are not parallel, $v + w$ is the diagonal of the parallelogram having the two vectors as sides (parallelogram rule). In physics the sum of vectors describes the composition of displacements and of forces.

Parallelogram rule:

u + v

is the diagonal of the parallelogram with sides

u

and

v

3. Multiplication by a scalar

Let $λ \in R$ and let $v$ be a vector. The vector $λ v$ has length equal to $∣ λ ∣$ times the length of $v$ , the same direction as $v$ , the same orientation if $λ > 0$ and the opposite orientation if $λ < 0$ ; if $λ = 0$ or $v = 0$ , one sets $λ v = 0$ . The vector $(- 1) v$ is denoted by $- v$ and is the opposite of $v$ ; the difference is $v - w = v + (- w)$ . If $v = A B$ and $w = A C$ , then $v - w = C B$ , that is, the difference is the vector going from the head of $w$ to the head of $v$ .

The difference

v - w

goes from the head of

w

to the head of

v

, because

w + (v - w) = v

Properties of the operations

For every triple of vectors $u, v, w$ and for every $λ, μ \in R$ the following properties hold.

Commutativity and associativity of the sum: $v + w = w + v$ and $(u + v) + w = u + (v + w)$ .
Identity element and opposite: $v + 0 = v$ and $v + (- v) = 0$ .
Distributivity: $λ (v + w) = λ v + λ w$ and $(λ + μ) v = λ v + μ v$ .
Mixed associativity and unit: $λ (μ v) = (λ μ) v$ and $1 v = v$ .

These are the same properties that define a vector space, and the general theory is covered in the topic on vector spaces.

Components in a Cartesian coordinate system

1. Components of a vector

Fix in space an orthogonal Cartesian coordinate system $O x y z$ and denote by $i$ , $j$ , $k$ the vectors of length $1$ oriented like the three axes. Every vector $v$ , applied at the origin, has its head at exactly one point $P = (v_{1}, v_{2}, v_{3})$ , and

v = O P = v_{1} i + v_{2} j + v_{3} k .

The numbers $v_{1}, v_{2}, v_{3}$ are the components of $v$ , and one writes $v = (v_{1}, v_{2}, v_{3})$ . In this way the vectors of space correspond to the triples of $R^{3}$ , and the vectors of the plane, with a coordinate system $O x y$ , to the pairs of $R^{2}$ . In particular $i = (1, 0, 0)$ , $j = (0, 1, 0)$ , $k = (0, 0, 1)$ and $0 = (0, 0, 0)$ .

In the plane

v = O P = 3 i + 2 j

, therefore

v = (3, 2)

2. Operations in components

Let $v = (v_{1}, v_{2}, v_{3})$ , $w = (w_{1}, w_{2}, w_{3})$ and $λ \in R$ . Then

v + w = (v_{1} + w_{1}, v_{2} + w_{2}, v_{3} + w_{3}), λ v = (λ v_{1}, λ v_{2}, λ v_{3}) .

The formula follows from the properties of the operations: adding $v_{1} i + v_{2} j + v_{3} k$ and $w_{1} i + w_{2} j + w_{3} k$ and collecting $i$ , $j$ , $k$ gives the components of the sum, and the same holds for multiplication by $λ$ . Geometric operations thus become computations with numbers; for $v = (1, 2, - 1)$ and $w = (3, 0, 2)$ one has $2 v - w = (2 - 3, 4 - 0, - 2 - 2) = (- 1, 4, - 4)$ .

3. Vector between two points and midpoint

Let $A = (a_{1}, a_{2}, a_{3})$ and $B = (b_{1}, b_{2}, b_{3})$ . Then

A B = (b_{1} - a_{1}, b_{2} - a_{2}, b_{3} - a_{3}),

and the midpoint of the segment $A B$ is

M = (\frac{a _{1} + b _{1}}{2}, \frac{a _{2} + b _{2}}{2}, \frac{a _{3} + b _{3}}{2}) .

By the head-to-tail rule $O A + A B = O B$ , hence $A B = O B - O A$ , which in components is the first formula. The midpoint satisfies $A M = \frac{1}{2} A B$ , therefore $O M = O A + \frac{1}{2} (O B - O A) = \frac{1}{2} (O A + O B)$ . By the same reasoning the point $P$ such that $A P = t A B$ has coordinates $A + t (B - A)$ ; for $t \in [0, 1]$ the point $P$ traces the segment $A B$ , for $t \in R$ the line $A B$ .

4. Linear combination

Given the vectors $v_{1}, \dots, v_{k}$ and the real numbers $λ_{1}, \dots, λ_{k}$ , the vector

λ_{1} v_{1} + λ_{2} v_{2} + \dots + λ_{k} v_{k}

is a linear combination of $v_{1}, \dots, v_{k}$ with coefficients $λ_{1}, \dots, λ_{k}$ . Writing a vector as a linear combination of given vectors means finding the coefficients, and equating the components yields a linear system. The expression $v = v_{1} i + v_{2} j + v_{3} k$ is the linear combination of the three unit vectors of the axes with coefficients equal to the components.

Coefficients of a combination

In the plane we look for a way to write $w = (5, 1)$ as a combination of $u = (1, 1)$ and $v = (1, - 1)$ . The equality $α u + β v = w$ , read component by component, is the system

{α + β = 5 α - β = 1

which has the unique solution $α = 3$ , $β = 2$ , therefore $w = 3 u + 2 v$ .

w = (5, 1)

is obtained by adding

3 u

and

2 v

with the head-to-tail rule

The theory continues with 6 more sections (magnitude distance and unit vectors, dot product and angle between vectors, orthogonality and orthogonal projection…), plus all worked examples, the solutions and the PDF.

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