Chapter 2

Vectors

We have discussed one fundamental use of $𝑛$ -tuples of real numbers: describing positions in space. But they also play a foundational role in the theory of vectors, which help us measure not locations but directions.

2.1Points and Free Vectors

Definition 2.1 (Vector). A vector is an object that stores both a length (called its magnitude) and a direction. We represent a vector by a directed line segment, or a little arrow in space.

This gives us an important distinction. A point records where something is; a vector records an amount and a direction. For example, the location of a car is a point, while its velocity is a vector.

There is one subtlety in our picture of a vector as an arrow. The location of the arrow is not part of the vector! If we slide an arrow across the page without changing its length or direction, the translated arrow represents the same vector. We call vectors free vectors to emphasize that they are not anchored at a particular point.

This is why the little arrows used to represent a single vector may appear in many different places. Each arrow has a tail and a head, but those belong to the drawing; the vector itself remembers only the displacement from the tail to the head.

Figure 2.1 Drag the arrowhead and the vector changes: its length or direction is now different. Drag the tail and the entire arrow slides without changing the vector. Turn on the other representatives to see the same free vector drawn in many different places.

Exercise 2.2 (Points or Vectors?). Which of the following quantities are positions? Which are vectors?

Where I parked my car.
The wind hitting me in the face.
The location of Mars in the solar system.
The velocity of Mars in the solar system.
Gravitational acceleration.

Exercise 2.3. Come up with some of your own scenarios that are measured using two-dimensional points, three-dimensional vectors, four-dimensional points, and four-dimensional vectors.

2.2Vector Components and Notation

To work with vectors, we need a means of writing them down using numbers. One idea is to use the same Cartesian coordinate system discussed in the previous chapter to express a vector in components.

Definition 2.4 (Vector Notation). To help avoid confusing points and vectors, we will use different notations for the two. For a point, we use a simple letter such as $𝑝$ , whereas for a vector we decorate the letter with an arrow, such as $⃗ 𝑢$ .

When using Cartesian coordinates, we write a point with round brackets,

𝑝 = (1, 2, 3),

whereas for a vector we use angle brackets,

⃗ 𝑢 = ⟨ 1, 2, 3 ⟩ .

We may also write a vector as a column of numbers when this saves space or makes a calculation easier to read:

⃗ 𝑢 = ⎛ ⎜ ⎜ ⎜ ⎝ 123 ⎞ ⎟ ⎟ ⎟ ⎠ .

The components of a vector measure how far the arrow travels in each coordinate direction. For example, $⟨ 3, - 1 ⟩$ travels three units in the positive $𝑥$ -direction and one unit in the negative $𝑦$ -direction. These components do not depend on where the representative arrow is drawn, which is the numerical observation made by the opening figure.

Definition 2.5 (The Zero Vector). The zero vector in an $𝑛$ -dimensional space is the vector with magnitude zero. In coordinates, it is the $𝑛$ -tuple of all zeros; in $ℝ 3$ it is $⃗ 0 = ⟨ 0, 0, 0 ⟩$ .

Geometrically, the zero vector is represented by an arrow whose head and tail coincide. It has no preferred direction, but it is needed for vector arithmetic in the same way that the number zero is needed for ordinary arithmetic.

2.3Arithmetic of Vectors

Vectors can be added and scaled. Both operations are easiest to understand geometrically first, and then write down in components.

Definition 2.6 (Vector Addition). Vector addition is defined to give the combined effect of two vectors. If $⃗ 𝑢$ and $⃗ 𝑣$ are vectors, then $⃗ 𝑢 + ⃗ 𝑣$ is defined geometrically by the diagonal of the parallelogram with sides $⃗ 𝑢$ and $⃗ 𝑣$ . This is equivalent to the vector formed by stacking $⃗ 𝑢$ and $⃗ 𝑣$ head-to-tail, in either order.

In Cartesian coordinates, vector addition is performed one component at a time:

⟨ 𝑢 1, 𝑢 2, 𝑢 3 ⟩ + ⟨ 𝑣 1, 𝑣 2, 𝑣 3 ⟩ = ⟨ 𝑢 1 + 𝑣 1, 𝑢 2 + 𝑣 2, 𝑢 3 + 𝑣 3 ⟩ .

The animation below shows why the parallelogram and head-to-tail constructions are really the same definition. Try walking the two vectors in both possible orders.

Figure 2.2 Following

⃗ 𝑢

and then

⃗ 𝑣

reaches the same endpoint as following

⃗ 𝑣

and then

⃗ 𝑢

. The diagonal of the parallelogram is their sum in either order.

Definition 2.7 (Scalar Multiplication). If $⃗ 𝑢$ is a vector and $𝑐$ is a number, the vector $𝑐 ⃗ 𝑢$ has magnitude $| 𝑐 | ‖ ⃗ 𝑢 ‖$ . It points in the same direction as $⃗ 𝑢$ when $𝑐 > 0$ and in the opposite direction when $𝑐 < 0$ . In Cartesian coordinates,

𝑐 ⟨ 𝑢 1, 𝑢 2, 𝑢 3 ⟩ = ⟨ 𝑐 𝑢 1, 𝑐 𝑢 2, 𝑐 𝑢 3 ⟩ .

Because we think of real numbers as being the kind of things that can scale vectors, we often call them scalars.

Multiplying by $- 1$ reverses a vector, giving its negative $- ⃗ 𝑢$ . This lets us define subtraction by

⃗ 𝑢 - ⃗ 𝑣 = ⃗ 𝑢 + (- ⃗ 𝑣),

which again means subtracting one component at a time.

Vector addition can be used to describe complicated motion in terms of simpler pieces. Indeed, this idea was used by the ancient Greeks in their planetary models, where complicated motions of objects in the heavens were modeled as combinations of circular motions, or epicycles.

The figure below shows two modern versions of the same idea. The Moon's position relative to the Sun is the sum of Earth's motion around the Sun and the Moon's motion around Earth. Mars's position as seen from Earth is a difference: its position relative to the Sun minus Earth's.

Figure 2.3 Two simple circular motions combine into a more complicated path. Addition builds the Moon's motion around the Sun; subtraction produces the apparent retrograde loops of Mars as viewed from Earth. The Moon's wobble is exaggerated so that it can be seen.

In these calculations, vector arithmetic works like the arithmetic of numbers, one coordinate at a time. It therefore satisfies many familiar algebraic laws.

Theorem 2.8 (Vector Arithmetic). Let $⃗ 𝑢, ⃗ 𝑣$ , and $⃗ 𝑤$ be vectors, and let $𝑐, 𝑘$ be scalars. Then

⃗ 𝑢 + ⃗ 𝑣 = ⃗ 𝑣 + ⃗ 𝑢, ⃗ 𝑢 + (⃗ 𝑣 + ⃗ 𝑤) = (⃗ 𝑢 + ⃗ 𝑣) + ⃗ 𝑤,

⃗ 𝑢 + ⃗ 0 = ⃗ 𝑢, ⃗ 𝑢 + (- ⃗ 𝑢) = ⃗ 0,

𝑐 (⃗ 𝑢 + ⃗ 𝑣) = 𝑐 ⃗ 𝑢 + 𝑐 ⃗ 𝑣, (𝑐 + 𝑘) ⃗ 𝑢 = 𝑐 ⃗ 𝑢 + 𝑘 ⃗ 𝑢,

(𝑐 𝑘) ⃗ 𝑢 = 𝑐 (𝑘 ⃗ 𝑢), 1 ⃗ 𝑢 = ⃗ 𝑢 .

2.4Displacement Between Points

One common use for vectors is to give directions to get from one point to another: that is, given points $𝑝$ and $𝑞$ in space, we want a vector starting at $𝑝$ and ending at $𝑞$ .

This vector encodes the magnitude and direction information of “if you are at $𝑝$ and you walk this amount in this direction, you will arrive at $𝑞$ .” We can construct such a vector by subtracting the coordinates of the starting point from the coordinates of the ending point.

Definition 2.9 (Vector from Two Points). The displacement vector from a point $𝑝 = (𝑝 1, 𝑝 2, 𝑝 3)$ to a point $𝑞 = (𝑞 1, 𝑞 2, 𝑞 3)$ is

⃗ 𝑑 = 𝑞 - 𝑝 = ⟨ 𝑞 1 - 𝑝 1, 𝑞 2 - 𝑝 2, 𝑞 3 - 𝑝 3 ⟩ .

The same definition works in every dimension.

Notice that the order matters: $𝑞 - 𝑝$ points from $𝑝$ to $𝑞$ , while $𝑝 - 𝑞$ points backward from $𝑞$ to $𝑝$ . The two displacement vectors are negatives of one another.

Figure 2.4 The arrow from

𝑝

𝑞

has components

𝑞 - 𝑝

. Translate the same arrow to the origin and turn on its distance triangle: the Pythagorean theorem from Chapter 1 now says

For example, let

𝑝 = (1, - 2, 3), 𝑞 = (5, 1, - 3) .

The displacement from $𝑝$ to $𝑞$ is

𝑞 - 𝑝 = ⟨ 5 - 1, 1 - (- 2), - 3 - 3 ⟩ = ⟨ 4, 3, - 6 ⟩ .

Its magnitude is

‖ 𝑞 - 𝑝 ‖ = \sqrt 42 + 32 + (- 6) 2 = \sqrt 61,

so the distance between the two points is $\sqrt 61$ .

2.5Magnitude and Unit Vectors

The vector $⟨ 𝑥, 𝑦, 𝑧 ⟩$ is the displacement vector from the origin to the point $(𝑥, 𝑦, 𝑧)$ . Its length, or magnitude, is the distance from the origin to its other endpoint, which we already know from the Pythagorean theorem.

Definition 2.10 (Magnitude). The magnitude of a vector $⃗ 𝑢 = ⟨ 𝑢 1, 𝑢 2, \dots, 𝑢 𝑛 ⟩$ is

‖ ⃗ 𝑢 ‖ = \sqrt 𝑢 21 + 𝑢 22 + \dots + 𝑢 2 𝑛 .

Thus, magnitude does not introduce a new distance formula. It gives the formula from Chapter 1 a new interpretation: the length of a displacement vector.

A vector of length 1 is called a unit vector. We think of these as measuring purely direction just as we think of numbers as measuring purely length.

Definition 2.11 (Unit Vector in a Given Direction). If $⃗ 𝑣$ is a nonzero vector, the unit vector in the direction of $⃗ 𝑣$ is denoted $ˆ 𝑣$ and is calculated by dividing $⃗ 𝑣$ by its own magnitude:

ˆ 𝑣 = 1 ‖ ⃗ 𝑣 ‖ ⃗ 𝑣 .

The hypothesis $⃗ 𝑣 \neq ⃗ 0$ is important: the zero vector has no direction, and dividing by its magnitude would mean dividing by zero.

Figure 2.5 Dividing

⃗ 𝑣

by its magnitude preserves its direction and produces a unit vector

ˆ 𝑣

. Multiplying

ˆ 𝑣

by a positive number

𝐿

then produces a vector of length

𝐿

in the same direction. A negative scalar reverses the direction.

Example 2.12 (A Unit Vector in a Given Direction). Find a unit vector in the direction of $⟨ 1, 2, 3, 4 ⟩$ .

First compute the magnitude:

‖ ⟨ 1, 2, 3, 4 ⟩ ‖ = \sqrt 12 + 22 + 32 + 42 = \sqrt 30 .

Dividing the vector by this magnitude gives

ˆ 𝑣 = 1 \sqrt 30 ⟨ 1, 2, 3, 4 ⟩ = ⟨ 1 \sqrt 30, 2 \sqrt 30, 3 \sqrt 30, 4 \sqrt 30 ⟩ .

Exercise 2.13 (A Vector of Prescribed Length). Find a vector of length $2$ in the direction of $⟨ 1, 1, 1, 1, 1, 1, 1, 1 ⟩$ .

Hint: first find a unit vector in this direction. What happens to its length if you scalar multiply it by $2$ ?

2.6Linear Combinations and Standard Bases

Vector addition and scalar multiplication can be used together to build new vectors from old ones.

Definition 2.14 (Linear Combination). A linear combination of a list of vectors is a new vector made by taking a sum of scalar multiples of the original vectors.

For example, if $⃗ 𝑢 = ⟨ 1, 2 ⟩$ and $⃗ 𝑣 = ⟨ 3, 4 ⟩$ , then

⃗ 𝑤 = 7 ⃗ 𝑢 - 2 ⃗ 𝑣 = 7 ⟨ 1, 2 ⟩ - 2 ⟨ 3, 4 ⟩ = ⟨ 7 - 6, 14 - 8 ⟩ = ⟨ 1, 6 ⟩ .

Cartesian coordinates are built from a collection of perpendicular axes. Each of these axes has a direction that we call a standard basis direction.

Definition 2.15 (Standard Basis). For $ℝ 2$ , the standard basis vectors are $⟨ 1, 0 ⟩$ and $⟨ 0, 1 ⟩$ , pointing along the positive directions of the $𝑥$ - and $𝑦$ -axes.

For $ℝ 3$ , the standard basis vectors are $⟨ 1, 0, 0 ⟩$ , $⟨ 0, 1, 0 ⟩$ , and $⟨ 0, 0, 1 ⟩$ , pointing along the positive directions of the $𝑥$ -, $𝑦$ -, and $𝑧$ -axes.

In general, the $𝑛$ standard basis vectors in $ℝ 𝑛$ have all zeros as components except for a single $1$ . The vector whose $1$ is in the $𝑖 t h$ component is called the $𝑖 t h$ standard basis vector.

For example,

⟨ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ⟩

is the fourth standard basis vector of twelve-dimensional space. In two and three dimensions, we give each basis vector a unique letter to aid readability instead of dealing with subscripts for only a handful of symbols.

Definition 2.16 (Standard Basis in $\RR^2$ and $\RR^3$). In $ℝ 2$ , we write

ˆ ı = ⟨ 1, 0 ⟩, ˆ ȷ = ⟨ 0, 1 ⟩ .

In $ℝ 3$ , we write

ˆ ı = ⟨ 1, 0, 0 ⟩, ˆ ȷ = ⟨ 0, 1, 0 ⟩, ˆ 𝑘 = ⟨ 0, 0, 1 ⟩ .

We can use these standard basis vectors to express any vector in space. For example, in $ℝ 3$ every vector is some amount in the $ˆ ı$ direction, some amount in the $ˆ ȷ$ direction, and some amount in the $ˆ 𝑘$ direction. This means we can write any vector as a linear combination of these:

⃗ 𝑢 = ⟨ 𝑥, 𝑦, 𝑧 ⟩ = 𝑥 ˆ ı + 𝑦 ˆ ȷ + 𝑧 ˆ 𝑘 .

The figure below begins with two arbitrary vectors $⃗ 𝑢$ and $⃗ 𝑣$ . Changing $𝑎$ and $𝑏$ changes the linear combination $𝑎 ⃗ 𝑢 + 𝑏 ⃗ 𝑣$ . After exploring this, switch to $ˆ ı$ and $ˆ ȷ$ and watch the coefficients $𝑎, 𝑏$ become the ordinary Cartesian components.

Figure 2.6 Nonparallel vectors in the plane can be combined to reach every point of the plane; parallel vectors can only reach points on one line. For the standard basis,

𝑎 ˆ ı + 𝑏 ˆ ȷ = ⟨ 𝑎, 𝑏 ⟩

, so the coefficients are exactly the components of the resulting vector.

Remark 2.17 (Vector Reference). For points $𝑝, 𝑞 \in ℝ 𝑛$ , vectors $⃗ 𝑢 = ⟨ 𝑢 1, \dots, 𝑢 𝑛 ⟩$ and $⃗ 𝑣 = ⟨ 𝑣 1, \dots, 𝑣 𝑛 ⟩$ , and a scalar $𝑐$ :

⃗ 𝑢 + ⃗ 𝑣 = ⟨ 𝑢 1 + 𝑣 1, \dots, 𝑢 𝑛 + 𝑣 𝑛 ⟩, 𝑐 ⃗ 𝑢 = ⟨ 𝑐 𝑢 1, \dots, 𝑐 𝑢 𝑛 ⟩, \to 𝑝 𝑞 = 𝑞 - 𝑝, d i s t (𝑝, 𝑞) = ‖ 𝑞 - 𝑝 ‖, ˆ 𝑣 = ⃗ 𝑣 ‖ ⃗ 𝑣 ‖ (⃗ 𝑣 \neq ⃗ 0) .

Vectors now give us a language for displacement, direction, and magnitude. But we still cannot measure the angle between two vectors, or answer a question like “how much of $⃗ 𝑣$ points in the direction of $⃗ 𝑢$ ?” Both questions will be answered by the dot product in the next chapter.