theadamabrams.com/live

Math 168818 November 2021

Warm-up: What have you learned about vectors from other classes?

This classComplex numbers

Rectangular formPolar form

PolynomialsFactoring

Irreducible polyn.Roots (zeros)

MultiplicitiesQuotient and remainder

VectorsMatrices

not linear algebra

linear algebra

Vectors as .ListsPointsArrows

The word “vector” can mean many things.At times we will think of a vector as

a list.a point.an arrow with its tail at the origin.an arrow with its tail anywhere.

There is another option:an element of an abstract vector space,

but we won’t use that idea of a vector in this class.

youtu.be/fNk_zzaMoSs (3B1B)a list. an arrow with its tail at the origin.an arrow with its tail anywhere.

an element of an abstract vector space,

Vectors as .ListsPointsArrows

A vector is a list of numbers.We can write the same list of numbers in many formats. For example,are all exactly the same vector.Each numbers is a component of the vector.

For , the “1st component” is , the “2nd component” is , etc.We often label the components with subscripts: . But sometimes we instead label a whole vector this way: and .

[5,3,8] 5 3u = ⟨u1, u2, u3⟩u1 u2

(5, 3, 8) [ 5 3 8 ]⟨5, 3, 8⟩

<latexit sha1_base64="x8bgif6wD7Spxg/zRsKO5Qc4Ixk=">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</latexit>

538

<latexit sha1_base64="TegcfVvri44rY9MTWk0xdi1yNy4=">AAACfnicbVFdb9MwFHUzPkb46sYjLxeqIUBK5lIK3VClCQTicYhmq9RUlePcZNYSJ7KdqVXU38Cv4RV+B/8Gpy0VTBzJ1tU599hX50ZlJrSh9FfL2blx89bt3Tvu3Xv3Hzxs7+2f6aJSHANeZIUaR0xjJiQGRpgMx6VClkcZnkeXHxr9/AqVFoUcmUWJ05ylUiSCM2OpWftFGGEqZB3lzCgxX0IfwhB6zTWAEGW8VWbtDvWPLHpH0PXpCkD9fo/SN6+3TIdscDrbaw0P4Mno4xhQ8iIWMoUhBKNP3gACKSyD7kYfffVKVaSK5bZjjhkzOLea99KDtfF47XvXUFa4EvkxaDSQ2JeHlUm8gRvGBa9ylIZnTOtJl5ZmWjNlBM9w6YaVxpLxS5bixJaS5ain9Sq+JRxYJoakUPZIAyv2b0fNcq0XeWQ7bRYX+rrWkP/TJnaywbQWsqyMHXT9UVJlYApodgGxUMhNtrAF40rYWYFfMMW4sRtzV4Fvc75e/An87JXf7fv0C+2cvN9Ev0sek6fkOemSt+SEfCanJCCcfCPfyQ/y0yHOM8dzDtetTmvjeUT+gTP4DcFFuNw=</latexit>

538

x y

z

(4,2,0)

(4,0,5)

(0,2,5)

Vectors as .Lists

PointsArrows

A vector is a point (a dot) in 2D or 3D space.

(4,2,5)

Vectors as .ListsPointsArrows

A vector is something that has a magnitude and a direction.In other words, it is an arrow.

x y

z

(4,2,0)

(4,0,5)

(0,2,5)

(4,2,5)

tip

tail

Vectors as .A vector is something that has a magnitude and a direction.In other words, it is an arrow.

ListsPointsArrows

Vectors as .A vector is something that has a magnitude and a direction.In other words, it is an arrow.

ListsPointsArrows

2 right

3 up

Vectors as .A vector is something that has a magnitude and a direction.In other words, it is an arrow.

ListsPointsArrows

Vectors as .A vector is something that has a magnitude and a direction.In other words, it is an arrow.

ListsPointsArrows

Vector variablesIn different text/videos, a vector variable might be written as any of these:

Often we use letters or for vectors.If the vector has a specific meaning, we might use a letter related to that meaning (for example, for a “normal vector”).

u, v, w a, b, c

n

u ⇀u u uu

Vector variablesTwo vectors are equal if their first components are equal and their second components are equal and so on.

Example: ⟨5, 1, 9⟩ = ⟨2+3, 66 , 13−4⟩

As with numbers, sometimes an equation describes one specific value

and sometimes there are many values that make an equation true:u = ⟨1, −3⟩

| u | = 5

x = −8

x2 − 4x + 3 = 0

OriginThe zero vector is in 2D and in 3D.Depending on context, a vector like might refer to any arrow that points in a direction right and up, orthe specific arrow from to , orthe point .

0 = ⟨0,0⟩ 0 = ⟨0,0,0⟩⟨5,1⟩

5 1(0,0) (5,1)

(5,1)

MagnitudeThe magnitude (or length or norm) of the vector is

.

In 2D or 3D, this is exactly the physical length of the arrow, or the length of the line segment from the origin to the point .

v = ⟨v1, v2, . . . , vn⟩

| v | = v21 + v2

2 + ⋯ + v2n

v

⟨4,2⟩

Magnitude

⟨4,2⟩

side length 4

side length 2

side length 20

The magnitude (or length or norm) of the vector is

.

In 2D or 3D, this is exactly the physical length of the arrow, or the length of the line segment from the origin to the point .

v = ⟨v1, v2, . . . , vn⟩

| v | = v21 + v2

2 + ⋯ + v2n

v

Scalar multiplicationFor this class, a scalar is a number (that is, not a vector).Given a scalar and a vector , we can multiply and to get

.Examples:

s v = ⟨v1, v2, . . . , vn⟩ sv

s v = ⟨sv1, sv2, . . . , svn⟩

3⟨8,1⟩ = ⟨24,3⟩12 ⟨8,1⟩ = ⟨4, 1

2 ⟩4⟨−3, 9.1⟩ = ⟨−12, 36.8⟩−2⟨5, −4⟩ = ⟨−10,8⟩0⟨5,7⟩ = ⟨0,0⟩

We say that is a scalar multiple of .

⟨24,3⟩⟨8,1⟩

Note that is not a scalar multiple of .

⟨24,10⟩⟨8,1⟩

Scalar multiplicationGeometrically, is a “stretched” version of .s v v

⟨2,1⟩ ⟨4,2⟩

⟨−1,−32

⟩

Parallel vectorsTwo vectors and are parallel if for some .

Some people require and say, for example, that and are “anti-parallel”. Some people call them parallel.

For any is a scalar multiple of , but is not parallel to .

u v u = s v s ≠ 0

s > 0 ⟨3,2⟩⟨−6, −4⟩

v , 0 v 0 v

Parallel vectorsTwo vectors and are parallel if for some .

Some people require and say, for example, that and are “anti-parallel”. Some people call them parallel.

For any is a scalar multiple of , but is not parallel to .

u v u = s v s ≠ 0

s > 0 ⟨3,2⟩⟨−6, −4⟩

v , 0 v 0 v

Vector additionAs lists, vectors are added by adding each coordinate.

Example: Example:

As arrows, vectors are added “tip-to-tail”.

⟨9, −4⟩ + ⟨5, 6⟩ = ⟨14, 2⟩<latexit sha1_base64="GhWcmTD5Qz/BaBwM7gVgAHbjsf8=">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</latexit>[58

]+

[−27

]=

[315

]

a b

Vector additionAs lists, vectors are added by adding each coordinate.

Example: Example:

As arrows, vectors are added “tip-to-tail”.

⟨9, −4⟩ + ⟨5, 6⟩ = ⟨14, 2⟩<latexit sha1_base64="GhWcmTD5Qz/BaBwM7gVgAHbjsf8=">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</latexit>[58

]+

[−27

]=

[315

]

a

b

a+b

Vector additionAs lists, vectors are added by adding each coordinate.

Example: Example:

As arrows, vectors are added “tip-to-tail”.

⟨9, −4⟩ + ⟨5, 6⟩ = ⟨14, 2⟩<latexit sha1_base64="GhWcmTD5Qz/BaBwM7gVgAHbjsf8=">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</latexit>[58

]+

[−27

]=

[315

]

a

b

a+b

u

vv

Vector additionAs lists, vectors are added by adding each coordinate.

Example: Example:

As arrows, vectors are added “tip-to-tail”.

⟨9, −4⟩ + ⟨5, 6⟩ = ⟨14, 2⟩<latexit sha1_base64="GhWcmTD5Qz/BaBwM7gVgAHbjsf8=">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</latexit>[58

]+

[−27

]=

[315

]

u

a

b

a+b

v

u+v

v

Vector additionAs lists, vectors are added by adding each coordinate.

Example: Example:

As arrows, vectors are added “tip-to-tail”.

⟨9, −4⟩ + ⟨5, 6⟩ = ⟨14, 2⟩<latexit sha1_base64="GhWcmTD5Qz/BaBwM7gVgAHbjsf8=">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</latexit>[58

]+

[−27

]=

[315

]

u

v

a

b

a+b v

u+v v+

u

Vector identitiesu + v = v + u

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

s(t v ) = (st) v

s( u + v ) = (s u ) + (s v )

(s + t) v = (s v ) + (t v )

u + 0 = u

u + (− u ) = 0

Vector identities

…

a + b = b + a

( a + b ) + c = a + ( b + c )

k(m a ) = (km) a

Basis vectorsLater, we will talk about the general idea of a “basis”, but for now we will use just one 2D example and one 3D example.

In 2D, the standard basis vectors are

and .

In 3D, the standard basis vectors are

and and .

ı = [10] 𝚥 = [0

1]ı = [

100] 𝚥 = [

010] k = [

001]

We can write any vector using scalar multiples, these basis vectors, and vector addition.

Basis vectorsWe can write any vector using scalar multiples, these basis vectors, and vector addition.

Examples:

[52] = [5

0] + [02] = 5 [1

0] + 2 [01] = 5 ı + 2 𝚥

[6

0.91−2 ] = 6 ı + 0.91 𝚥 − 2 k

[401] = 4 ı + k

[ab] = a ı + b 𝚥

[520] = 5 ı + 2 𝚥

Vector subtractionusing coordinates

geometrically

Vector subtractionWe can subtract vectors using coordinates.

Example: Example:

⟨9, −4⟩ − ⟨5, 6⟩ = ⟨4, −10⟩<latexit sha1_base64="ImSGXTrsg3awIUC6x9Tr2ZpxD6U=">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</latexit>[58

]−[−27

]=

[71

]

geometrically

Vector subtractionWe can subtract vectors using coordinates.

Example: Example:

What does mean geometrically?We could first find the scalar multiple and then use tip-to-tail addition to find .

⟨9, −4⟩ − ⟨5, 6⟩ = ⟨4, −10⟩

u − v− v = (−1) v

u + (− v )

<latexit sha1_base64="ImSGXTrsg3awIUC6x9Tr2ZpxD6U=">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</latexit>[58

]−[−27

]=

[71

]

What does mean for numbers?a − b

New meanings for old thingsWhat does mean?

More advanced: no pictures, just .

What does mean?

5 × 3

5 + 5 + 5

5 × 13 ? ?5 × 9.2 7.65 × (−12)

Multiplication can have different meanings or interpretations.This is also true for subtraction.

What does mean on a number line?

Answer: The number describes how to move from to

5 − 3

5 − 3 3 5.

Subtraction

3 5

+2

−2 4

+6

−4 −1

+3

In general, describes how to move from to .

b − aa b

SubtractionWhat does mean on a number line?

Answer: The number describes how to move from to

5 − 3

5 − 3 3 5.

In general, describes how to move from to .

b − aa b

3 5

-2

To go from to instead, we move left, which is why is negative.5 3 3 − 5

Vector subtractionThe vector points from the tip of to the tip of .u − v v u

u

v

u-v

This agrees with finding by adding tip-to-tail.u − v u + (− v )

u

v

In general, describes how to move from to .

b − aa b

Note: The tails (start) of u and v must be at the same place to use this method.

Vector subtractionThe vector points from the tip of to the tip of .u − v v u

uv

u

v

u-v

This agrees with finding by adding tip-to-tail.u − v u + (− v )-v

u+(-v)

Note: The tails (start) of u and v must be at the same place to use this method.

u

v

u-v

Vector points from the tip of to the tip of .

u − v vuNote: The tails (start) of u and v must be at the same place to use this method.

Multiplication of vectorsCombining and into is not actually useful. We will never do this.Instead, we have “dot product” and “cross product” of vectors.

can be done for vectors of any dimension.

will only be done in 3D.

We will never write without either or .

a = ⟨a1, a2⟩ b = ⟨b1, b2⟩ ⟨a1b1, a2b2⟩

a ⋅ ba × b

a b ⋅ ×

Dot productThe dot product (or inner product or scalar product) of

and is

.

This is a number, not a vector.Examples:

a = ⟨a1, a2, . . . , an⟩ b = ⟨b1, . . . , bn⟩a ⋅ b = a1b1 + a2b2 + ⋯ + anbn

⟨5,7⟩ ⋅ ⟨8,2⟩ = 40 + 14 = 54⟨3, −1,8⟩ ⋅ ⟨0,4,2⟩ = 0 + −8 + 16 = 8⟨x, 6⟩ ⋅ ⟨5,2⟩ = 5x + 12⟨2, −1⟩ ⋅ ⟨t − 9,t⟩ = 2(t − 9) − t = t − 18

Dot productThe dot product has another formula:

where is the angle between and .

Using both of these formulas, we can find the angle between vectors.

a ⋅ b = | a | | b |cos θθ a b

b

a

θ

Example: Find the acute angle between .⟨ 3,1⟩ and ⟨0,7⟩a b

| | = =

| | = =

= ( )(0) + (1)(7) = 7 Since is also (2)(7)cos , we know (2)(7)cos = 7 → cos = 1/2 → = 60°

a 3 3+1 = 4 = 2

b 49 = 7

a ⋅ b 3a ⋅ b θθ θ θ

( )2 + (1)2

(7)2 + (0)2

Example: Find the acute angle between .⟨4,3⟩ and ⟨1,6⟩

| | = | | =

= (4)(1) + (2)(6) = 4 + 12 = 16 So 5 cos = 16 cos = cannot be solved by hand.

We say .

a 16+9 = 25 = 5b 1+36 = 37

a ⋅ b37 θ

θ 165 37θ = arccos( 16

5 37 ) or cos-1( 165 37 )