Vectors in R 2 and R3 by Ryan C. Daileda, Lecture notes of Calculus

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Vectors in R^2 and R^3

Ryan C. Daileda

Trinity University

Calculus III

Introduction

Vectors represent quantities that have both size (magnitude) and direction.

Examples of vector quantities: displacement, velocity, acceleration, force, electric/magnetic fields, etc.

In the context of vectors, real numbers are referred to as scalars.

Examples of scalar quantities: speed, mass, temperature, energy, etc.

Remarks

We use boldface type for variables that represent vectors, e.g. u, v, w.

Alternate terminology:

magnitude = length

terminal point = tip

initial point = tail

The magnitude of a vector v is denoted |v|.

There’s no standard way to denote the direction of a vector.

Vector Equality

Two vectors u and v are considered equal (i.e. u = v) whenever they have the same magnitude and direction. Equality of vectors does not depend on their location in space.

v

u

v

u

v

u

u = v u 6 = v u 6 = v

Vector Arithmetic

Vector Addition: Two vectors can be added by positioning them “tip-to-tail,” and then inserting a vector from the tail of the first to the tip of the second.

u

v

tip-to-tail u v

u + v

The following diagram shows that vector addition is commutative:

u + v = v + u.

u v

u + (^) v

v u

It is also geometrically clear that the zero vector “acts like zero”:

v + 0 = 0 + v = v.

v

(^2) v

2

(^1) _ (^) v^ (-1)^ v^ = - v^^2

-^3 _^ v

Notice that | 0 v| = | 0 | · |v| = 0 · |v| = 0, which means that

0 v = 0.

So again zero “acts like zero.”

Vector Subtraction:

We combine addition and scalar multiplication and define

u − v = u + (−v).

Because −v interchanges the tip and tail of v, we have the following diagram:

v u

u - v

That is, we subtract vectors by placing them “tail-to-tail.”

Notice that, in particular, this means v − v = 0.

Vector Components

We now consider vectors from the analytic point of view.

Given a vector v (in R^2 or R^3 ) we position it with its tail at the origin:

v (^) move tail v to origin O

P =( a,b,c )

In this configuration, the coordinates of the tip of v (the point P) are the components of v.

We write v = 〈a, b, c〉.

Remarks

Both points and vectors now have “coordinates.” The difference between the two is primarily how we interpret them:

• The coordinates of a point tell you its location.

• The components of a vector tell you where the tip is, relative to the tail. Vector components provide a symbolic means of talking about magnitude and direction.

We will focus on components in R^3 , but everything that follows holds for R^2 as well, if one simply “forgets” the third component.

Examples

Example 1 If a = 〈 1 , 2 − 3 〉 and b = 〈− 2 , − 1 , 5 〉, find a + b, 2a + 3b and |a − b|.

Solution. We have

a + b = 〈1 + (−2), 2 + (−1), − 3 + 5〉 = 〈− 1 , 1 , 2 〉.

Likewise

2 a + 3b = 〈 2 , 4 , − 6 〉 + 〈− 6 , − 3 , 15 〉 = 〈− 4 , 1 , 9 〉.

Finally

|a − b| =

32 + 3^2 + (−8)^2 =

Example 2

Find the components of the vector that points from A = (a 1 , a 2 , a 3 ) to B = (b 1 , b 2 , b 3 ).

Solution. We introduce the origin O and consider the following diagram:

b

O

B

A

a

AB

Example 3

Find a unit vector (vector of length 1) that points in the direction of v = 〈− 1 , 3 , 4 〉.

Solution. Notice that the vector (^) |^1 v| v has magnitude

∣∣ ∣ ∣

|v|

v

∣ =^

|v|

· |v| = 1,

and the same direction as v since 1/|v| > 0.

So we simply need to “divide” v by |v| to obtain the vector we want:

u =

(−1)^2 + 3^2 + 4^2

The Standard Basis

The standard basis vectors are

i = 〈 1 , 0 , 0 〉, j = 〈 0 , 1 , 0 〉, k = 〈 0 , 0 , 1 〉.

They represent unit displacements along the coordinate axes.

Every vector can be expressed in terms of i, j and k:

〈a, b, c〉 = 〈a, 0 , 0 〉 + 〈 0 , b, 0 〉 + 〈 0 , 0 , c〉

= a〈 1 , 0 , 0 〉 + b〈 0 , 1 , 0 〉 + c〈 0 , 0 , 1 〉

= ai + bj + ck.

This alternate notation will be useful when we discuss the cross product of vectors.