MATH 19520/51 Class 1

Minh-Tam Trinh

University of Chicago

2017-09-25

Minh-Tam Trinh MATH 19520/51 Class 1

Before beginning class proper, we will discuss course logistics andschedule the problem session.

0 Background notation and terminology.1 Coordinate systems and the “right-hand rule” convention(RHR).

2 Vectors and their magnitude.3 Linear combination of vectors.4 Dot product and its properties.

Minh-Tam Trinh MATH 19520/51 Class 1

Background

x ∈ X “x is an element of the set X ,” or“x is contained in X ”

x < X “x is not an element of the set X ”

{x ∈ X : x satisfies P } the subset of X of elements that satisfyproperty P

X 2 the set of ordered pairs of elements of XX n the set of ordered n-tuples of elements of X

{x1, x2, x3} unordered set(x1, x2, x3) ordered tuple

R the set of real numbers

Minh-Tam Trinh MATH 19520/51 Class 1

Coordinate Systems

Coordinate space in n dimensions is denoted Rn. The elements ofRn are called n-vectors, or just vectors, and take the form

®u = (u1, u2, . . . , un),(1)

where ui ∈ R for all i. (Stewart writes 〈u1, u2, . . . , un〉.)

The element (0, 0, . . . , 0) is called the zero vector and denoted ®0.

Geometric interpretation of a vector:1 Absolute: As a position in space.2 Relative: As a displacement with fixed direction and

magnitude.

When we interpret ®0 as a position, it is called the origin.

Minh-Tam Trinh MATH 19520/51 Class 1

Coordinate Systems

Coordinate space in n dimensions is denoted Rn. The elements ofRn are called n-vectors, or just vectors, and take the form

®u = (u1, u2, . . . , un),(1)

where ui ∈ R for all i. (Stewart writes 〈u1, u2, . . . , un〉.)

The element (0, 0, . . . , 0) is called the zero vector and denoted ®0.

Geometric interpretation of a vector:1 Absolute: As a position in space.2 Relative: As a displacement with fixed direction and

magnitude.

When we interpret ®0 as a position, it is called the origin.

Minh-Tam Trinh MATH 19520/51 Class 1

Coordinate Systems

Coordinate space in n dimensions is denoted Rn. The elements ofRn are called n-vectors, or just vectors, and take the form

®u = (u1, u2, . . . , un),(1)

where ui ∈ R for all i. (Stewart writes 〈u1, u2, . . . , un〉.)

The element (0, 0, . . . , 0) is called the zero vector and denoted ®0.

Geometric interpretation of a vector:1 Absolute: As a position in space.2 Relative: As a displacement with fixed direction and

magnitude.

When we interpret ®0 as a position, it is called the origin.

Minh-Tam Trinh MATH 19520/51 Class 1

Potential ambiguity when drawing R3. Two possibilities, up torotating the axes:

(2)

z

yx

or

z

xy

(In both figures, (1, 1, 1) is meant to point out of the page.)

Minh-Tam Trinh MATH 19520/51 Class 1

The right-hand rule (RHR) convention: We always choose

(3)

z

yx

Stick the thumb of your right hand in the positive z-direction. Theremaining fingers on your right hand curl from the positivex-direction toward the positive y-direction.

Minh-Tam Trinh MATH 19520/51 Class 1

Magnitude

The magnitude or length of the vector ®u = (u1, . . . , un) is

|®u| =√u21 + · · · + u

2n.(4)

Magnitude is to vectors as absolute value is to real numbers: Ittells us the distance from the origin.

Example

If a box has side lengths 1 m × 2 m × 3 m, then the farthestdistance between two corners is

√12 + 22 + 32 =

√14 m.

Minh-Tam Trinh MATH 19520/51 Class 1

Magnitude

The magnitude or length of the vector ®u = (u1, . . . , un) is

|®u| =√u21 + · · · + u

2n.(4)

Magnitude is to vectors as absolute value is to real numbers: Ittells us the distance from the origin.

Example

If a box has side lengths 1 m × 2 m × 3 m, then the farthestdistance between two corners is

√12 + 22 + 32 =

√14 m.

Minh-Tam Trinh MATH 19520/51 Class 1

Linear Combinations

Vectors in Rn can be added to one another and rescaled by realnumbers. Real numbers are also called scalars.

If ®u = (u1, . . . , un) and ®v = (v1, . . . , vn), then

®u + ®v = (u1 + v1, . . . , un + vn).(5)

If a ∈ R, then

a®u = (au1, . . . , aun).(6)

The inverse of ®u is the vector –®u = –1®u, the opposite displacementfrom ®u. By definition, ®u – ®v = ®u + (–®v).

Minh-Tam Trinh MATH 19520/51 Class 1

Linear Combinations

Vectors in Rn can be added to one another and rescaled by realnumbers. Real numbers are also called scalars.

If ®u = (u1, . . . , un) and ®v = (v1, . . . , vn), then

®u + ®v = (u1 + v1, . . . , un + vn).(5)

If a ∈ R, then

a®u = (au1, . . . , aun).(6)

The inverse of ®u is the vector –®u = –1®u, the opposite displacementfrom ®u. By definition, ®u – ®v = ®u + (–®v).

Minh-Tam Trinh MATH 19520/51 Class 1

Linear Combinations

Vectors in Rn can be added to one another and rescaled by realnumbers. Real numbers are also called scalars.

If ®u = (u1, . . . , un) and ®v = (v1, . . . , vn), then

®u + ®v = (u1 + v1, . . . , un + vn).(5)

If a ∈ R, then

a®u = (au1, . . . , aun).(6)

The inverse of ®u is the vector –®u = –1®u, the opposite displacementfrom ®u. By definition, ®u – ®v = ®u + (–®v).

Minh-Tam Trinh MATH 19520/51 Class 1

Example

Consider vectors ®u = (0, 2) and ®v = (1, –2). The set of vectors

{®u + t®v ∈ R2 : t ∈ R}(7)

corresponds to a line in the (x, y)-plane. If we interpret t as time,then this line passes through (0, 2) at time t = 0.

In fact, the equation of this line is y = –2x + 2. Check this!

Minh-Tam Trinh MATH 19520/51 Class 1

Example

Consider vectors ®u = (0, 2) and ®v = (1, –2). The set of vectors

{®u + t®v ∈ R2 : t ∈ R}(7)

corresponds to a line in the (x, y)-plane. If we interpret t as time,then this line passes through (0, 2) at time t = 0.

In fact, the equation of this line is y = –2x + 2. Check this!

Minh-Tam Trinh MATH 19520/51 Class 1

In the number line R, the distance between numbers a and b is theabsolute value of their di�erence: |a – b|.

In Rn, the distance between ®u and ®v is the magnitude of theirdi�erence:

|®u – ®v| =√(u1 – v1)2 + · · · + (un – vn)2.(8)

Geometrically, this is the distance between the correspondingpositions in n-dimensional space.

Minh-Tam Trinh MATH 19520/51 Class 1

In the number line R, the distance between numbers a and b is theabsolute value of their di�erence: |a – b|.

In Rn, the distance between ®u and ®v is the magnitude of theirdi�erence:

|®u – ®v| =√(u1 – v1)2 + · · · + (un – vn)2.(8)

Geometrically, this is the distance between the correspondingpositions in n-dimensional space.

Minh-Tam Trinh MATH 19520/51 Class 1

Example

Fix a vector (a, b) ∈ R2 and a real number r > 0. Then the points(x, y) at distance r from (a, b) are the solutions of the equation

(x – a)2 + (y – b)2 = r2.(9)

E.g., the equation of the unit circle at the origin is x2 + y2 = 1.

Example

Similarly, in R3, the points (x, y, z) at distance r from (a, b, c) arethe solutions of the equation

(x – a)2 + (y – b)2 + (z – c)2 = r2.(10)

Minh-Tam Trinh MATH 19520/51 Class 1

Example

Fix a vector (a, b) ∈ R2 and a real number r > 0. Then the points(x, y) at distance r from (a, b) are the solutions of the equation

(x – a)2 + (y – b)2 = r2.(9)

E.g., the equation of the unit circle at the origin is x2 + y2 = 1.

Example

Similarly, in R3, the points (x, y, z) at distance r from (a, b, c) arethe solutions of the equation

(x – a)2 + (y – b)2 + (z – c)2 = r2.(10)

Minh-Tam Trinh MATH 19520/51 Class 1

Example

Geometrically, what is the set of (x, y, z) ∈ R3 that satisfy

x2 + y2 = z2?(11)

If we fix a value of z, then the set of possibilities for (x, y) forms acircle of radius |z|. As |z| gets larger, the radius gets larger at thesame rate.

Ultimately, the solution set is a pair of cones, their tips meeting atthe origin. Draw it!

Minh-Tam Trinh MATH 19520/51 Class 1

Example

Geometrically, what is the set of (x, y, z) ∈ R3 that satisfy

x2 + y2 = z2?(11)

If we fix a value of z, then the set of possibilities for (x, y) forms acircle of radius |z|. As |z| gets larger, the radius gets larger at thesame rate.

Ultimately, the solution set is a pair of cones, their tips meeting atthe origin. Draw it!

Minh-Tam Trinh MATH 19520/51 Class 1

Dot Product

The dot product of ®u and ®v is

®u · ®v = u1v1 + · · · + unvn.(12)

As we’ll see, it depends on both the magnitudes of ®u, ®v and howclosely their directions align.

Warning!

The dot product of two vectors is not a vector but a scalar.

Minh-Tam Trinh MATH 19520/51 Class 1

Dot Product

The dot product of ®u and ®v is

®u · ®v = u1v1 + · · · + unvn.(12)

As we’ll see, it depends on both the magnitudes of ®u, ®v and howclosely their directions align.

Warning!

The dot product of two vectors is not a vector but a scalar.

Minh-Tam Trinh MATH 19520/51 Class 1

Example

If kumquats, lychees, mangosteens are priced at $2, $8, $14 per lb,respectively, then the dot product

(k lbs,ℓ lbs,m lbs) · ($2/lb, $8/lb, $14/lb) = $(2k + 8ℓ + 14m)(13)

is the price of buying k lbs of kumquats, ℓ lbs of lychees, and m lbsof mangosteens.

Minh-Tam Trinh MATH 19520/51 Class 1

In some ways, dot product of vectors behaves like multiplication ofnumbers.

It is commutative:

®u · ®v = ®v · ®u.(14)

It distributes over addition of vectors:

®u · (®v + ®w) = ®u · ®v + ®u · ®w.(15)

However, since it yields a number not a vector, expressions like®u · ®v · ®w don’t make sense!

Minh-Tam Trinh MATH 19520/51 Class 1

In some ways, dot product of vectors behaves like multiplication ofnumbers.

It is commutative:

®u · ®v = ®v · ®u.(14)

It distributes over addition of vectors:

®u · (®v + ®w) = ®u · ®v + ®u · ®w.(15)

However, since it yields a number not a vector, expressions like®u · ®v · ®w don’t make sense!

Minh-Tam Trinh MATH 19520/51 Class 1