vectors - 3.1, 3.2, 3.3, 3.4, 3.5

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Section Four: Vectors

Textbook: Ch. 3.1, 3.2, 3.3, 3.4, (3.5)

GOALS OF THIS CHAPTER- revisit the idea of a vector and give more detail to this notion

-basic computation with vectors in 2-space: length, addition, scalarmultiplication, dot product, projections, angles between vectors, unitvectors

-basic computation with vectors in N-space: length, addition, scalarmultiplication, dot product, cross product, projections, anglesbetween vectors, unit vectors

-see the properties of the dot product and crossproduct

-Cauchy-Schwarz and Triangle Inequalities

-equations of lines and planes (if time permits)

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(RE)INTRODUCTION

We have seen vectors but just what is a vector? There are twostandard definitions:

We will still represent a vector as a bold-faced lowercase letter, or as an underlined lower case letter.

The geometric definition of a vector is the set of all directed linesegments equivalent to a given directed line segment.

The algebraic definition of an n-vector is the n-tuple (x1, x2, , xn).

Each xi is called a component or an entry of the vector.

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(RE)INTRODUCTION

For starters, we will work in the xy-plane (or in 2-

space) and then we will extend ideas to N-space.

Every vector in 2-space has two components.

2

1=u =

We will represent a vector as a column matrix, or as an orderedn-tuple.

2 is the first

component of thisvector

(2, 1)

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(RE)INTRODUCTION

Geometrically, we view vectors as directed linesegments that have both an initial endand aterminal end.

Initial End

Terminal End

This may not always the case (for future mathclasses); it really depends on what co-ordinate

system you are working in.

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(RE)INTRODUCTION

x

y

2

1

=u =

(2,1)

In the xy-plane, we can view a vectoras a directed line segment starting atthe origin and heading towards acertain point.

u

(2, 1)

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(RE)INTRODUCTION

x

y

2

1

=u =

(2,1)

The angle that the vector makes withthe x-axis is the directionof thevector.

u

(2, 1)

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(RE)INTRODUCTION

x

y

2

1

=u =

(2,1)

The magnitudeof this line segment isits length, or how long it is.

u

magnitude

(2, 1)

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(RE)INTRODUCTION

So for every vector, we can create a line segmentstarting at the origin. Also, for every line segmentstarting at the origin, we can create a vector. Butwhat happens if our line segment does not start atthe origin?

(2,2)

v

x

y

(4,3)

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(RE)INTRODUCTION

Subtract the initial end from the terminal end!

(2,2)

v

x

y

(4,3)

4 - 2

3 - 2

= =v2

1

x2 x1

y2 y1

=

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(RE)INTRODUCTION

I often refer to this as centering the vector. Nowyou can work with a vector at the origin!

x

y

=v 2

1

2

1

=

uu

But wait, u and v were indifferent spots and are both

equal?

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(RE)INTRODUCTION

So it doesnt really matter where this vector is inthe xy-plane, it is still treated as (2, 1). This meansvectors can be equal and not overlap one another!

v

x

y

=v 2

1

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CALCULATING MAGNITUDE

To calculate the magnitude of a vector, we need toget at its length. We denote the length with twovertical bars (looks like absolute value but it doesnot mean absolute value!): |v|. Lets say we have avector at the origin:

v

x

y

(a, b)

b

a

a2+b2Using Pythagoras Theorem,we have found the length of avector v= (a, b) to be

|v| = a2+b2

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Ex. 1 Length of a vector not at origin

Let the points P(1,2) and Q(-4,6) be the initial pointand terminal point of a vector. What is the length ofthis vector?

|PQ| = (-5)2+(4)2

First, center the vector at the origin!PQ = (-41, 6-2) = (-5, 4)

Now you can use Pythagoras Theorem:

= 41So the length ofthe vector PQ issquare root of

thirty-one.

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VECTOR ADDITION

To add vectors it is as easy as just adding thecorresponding components.

ux1

y1= v

x2

y2=

u + vx1 + x2

y1 + y2

=x1

y1

x2

y2

=+

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VECTOR ADDITION

Geometrically, vectors in 2-space follow theparallelogram law of addition.

v

x

y

v

u u

u + v

Remember that we can movethe vector anywhere in thexy-plane and not change it.The sum u+vends up as thediagonal of a parallelogram.

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VECTOR ADDITION

Ex. 2 Adding vectors

Find the vector u + vif

-3

1

=u =v

-3

1

=u + v +4

-2

1

-1

=-3+4

1+(-2)

=

You are allowed to switch it around to make it work!

(4, -2)

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VECTOR SCALAR MULTIPLICATION

Scalar multiplication occurs when we want tostretch the vector by a certain amount. In the xy-plane this has the effect of multiplying allcomponents by a common number.

ux1

y1=

cu =x1

y1

c =cx1

cy1

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Geometrically, vectors in 2-space will stretch orreflect (or both) when being scalar multiplied.

v

x

y

2v

-v

What if we scalarmultiply by zero? Well,

then we get the zerovector, or 0 = (0, 0).This is the only vectorwith a magnitude ofzero and any direction

we choose.

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If u = (2, -3) and v= (-1, -1), then find the vector u 3v.

Ex. 3 Mixing it up

-u 3v= -(2, -3) -3(-1, -1)

= (-2, 3) + (3, 3)

= (-2+3, 3+3)

= (1, 6)

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DOT PRODUCT AND ANGLES

The dot product is a way to take two vectors andreturn a single number. The dot product gives usinformation about the angle between the twovectors.

ux1

y1=

uv = x2

y2

v

x2

y2=

= x1x2 + y1y2X1

y1

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It turns out that we can couple the dot productwith the magnitudes of the vectors to get at thecosine of the angle between the vectors (you haveto use the cosine law to get here):

uvcos() = |u| |v|

x

y

v

u

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Lets take a closer look at this formula.

uvcos() =

|u| |v|

Cosine fluctuates betweenthe y-values 1 and -1. In fact,1, -1, and 0 are the threespecial numbers we will lookat.

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What happens if cos() = -1? This happens at = 180. This means the vectors are parallel butfacing opposite directions.

= 180 v

u

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What happens if cos() = 1? This happens at = 0 or 360. This means the vectors are paralleland lie on top of one another.

= 0 or 360vu

Oh baby, you cancos()=1 me any

day!

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What happens if cos() = 0? This happens at = 90. This means the vectors are perpendicular toone another. When two vectors meet at a 90 degreeangle, we call them orthogonal. This only happenswhen uv= 0

= 90 v

u

uvcos() =

|u| |v|

uv0 =

|u| |v|

uv0 =

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Ex. 4 Determining if two vectors are orthogonal

Given the vectors u = (2, -3) and v= (-1, -1),determine if they are orthogonal.

u v= -1

-1

= -2 +3 = 1

So these vectors are not orthogonal. If the dotproduct was zero, then they would be orthogonal.

2

-3

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Ex. 5 Determining angles between vectors

Given the vectors u = (2, -3) and v= (-1, -1),determine the cosine of the angle between them.

You will never be asked to solve for unless youget one of the three special values of cos().

uvcos() =|u| |v|

= 14+9 1+1

= 126

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VECTOR PROJECTIONS

For two vectors u and v, we can break down u into acomponent that is parallel to vand a component thatis perpendicular to v.

v

u

component of uparallel to v

component of uperpendicular to v

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VECTOR PROJECTIONS

These components of u are used in higher algebrawhen trying to find orthonormal bases(we will notlearn this but you can look on 395 of text for more).

The component of u parallel to v is often called theprojection of u on v, or projvu. Think of it like usshadow on v.

v

u

proj vu


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VECTOR PROJECTIONS

To calculate projvu, we can use the following:

v

u

proj vu


uv

|v|2vproj vu =

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VECTOR PROJECTIONS

The component of u perpendicular to vis oftenwritten as the following:

v

u

proj vu

(uv)u -

|v|2v

(uv)u -

|v|2v

This formula comes from the vector addition:

u = component parallel to v+ component perpendicular to vu = projvu + component perpendicular to vu projvu = component perpendicular to v

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VECTOR PROJECTIONS

Finally, we have a scalar number that tells us howlong u is in the direction of v. It is often called the

component ofuin the direction ofv(dont get itconfused with the other two components, which arevectors!).

v

u

uv

|v|

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VECTOR PROJECTIONS

Ex. 6 Creating a Projection- done in class

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UNIT VECTORS

Unit vectorsare special vectors that have amagnitude of one. If you calculate the length of a

vector and get one, then it is a unit vector.

What if the length of a vector is not equal to one? It is possible tocreate unit vectors in the same direction or the opposite directionwith existing vectors.

ux1

y1

=

ux1

y1

=1

|u|

Unit vectors aregiven a hat.

You can choose the +1 or the -1.Using +1 gives a unit vector inthe same direction as u andusing -1 gives a unit vector inthe opposite direction.

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UNIT VECTORS

There are two special unit vectors: one that runsalong the x-axis (1, 0) and one that runs along the y-

axis (0, 1). It is possible to show that these vectorscapture all the information of 2-space.

That is, a basisfor 2-space are the vectors (1, 0)and (0, 1). You will see more of this in second year

algebra courses.

x

y

i

j

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UNIT VECTORS

One of the things we can do is show that any vectorin 2-space is a linear combinationof these two unit

vectors.

x

y

i

j

v

So if we use vector addition,we have shown thatv = i + 2j

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Ex. 7 Unit vectors-done in class

Ex. 8 Linear combinations-done in class

UNIT VECTORS

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INTRODUCTION TO N-SPACE

What we want to do is generalize 2-space to higherdimensions. In the next slides, we will work in the

xyz-plane (or in higher coordinate axes). Most of thetime it is impossible to visualize these complexspaces.

Every vector in n-space has n components. Every vector in n-spacewill still be represented as a column matrix or an ordered n-tuple.

u1

u2

un

=u =

u2 is the secondcomponent of this

vector(u1, u2, , un)

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The set of all vectors with n components is called n-space. We wont be plotting vectors in n-space, but

they would still look like directed line segments.

This means we can still connect points in n-space and referto the initial end or terminal end of a vector.

If P(x1, x2, , xn) and Q(y1, y2, , yn) are two points in n-space, then:

PQ =

QP =

Vector from P to Q

Vector from Q to P

INTRODUCTION TO N-SPACE

(y1-x

1, y

2-x

2, , y

n-x

n)

(x1-y1, x2-y2, , xn-yn)

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|u| = (u1)2+(u2)2++(un)2


The magnitude |u| of a vector u in n-space is givenby the generalized Pythagoras Theorem. Provided

your vector is at the origin:

u1

u2

un

=u

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Ex. 9 Length of a vector in n-space

What is the length of the vector y = (1, 0, -2)?

|y| = (1)2+(0)2 +(-2)2

= 5

So the length of the vector y is the square rootof five.

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VECTOR ADDITION

Similar to 2-space, the addition of two vectors in n-space is defined below.

u =

v=

u + v =

=

(u1, u2, , un)

(v1, v2, , vn)

(u1, u2, , un) + (v1, v2, , vn)

(u1 + v1, u2 + v2, , un + vn)

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Vectors in n-space still stretch and/or reflect whenscalar multiplied. We still multiply all components by

a common number:

cu = c

u =

=

(u1, u2, , un)

(u1, u2, , un) (cu1, cu2, , cun)

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PROPERTIES OF ADDITION ANDSCALAR MULTIPLICATION

Thm. 10 Properties of Vector Addition and Scalar Multiplication-done in class

Proof-done in class

Ex. 11 Addition and Scalar Multiplication-done in class

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Just like in 2-space, the dot product returns a singlenumber that gives us some information about the angle

between the two vectors. Sometimes the n-space dotproduct is called the standard inner product.

uv = = u1v

1+ u

2v

2+ + u

nv

n

u =

v =

v1

v2

vn

(u1, u2, , un)

(v1, v2, , vn)

u1

u2

un

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In n-space, the cosine of the angle between twovectors is the same as it is in 2-space:

uvcos() =

|u| |v|

-two vectors are still perpendicular, or orthogonal,if their dot product is zero

-two vectors are still parallel and facing the samedirection if cos() = 1

-two vectors are still parallel and facing oppositedirection if cos() = -1

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Thm. 12 Properties of the N-space Dot Product-done in class

Proof-done in class

Ex. 13 Dot Products and Angles-done in class


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UNIT VECTORS

Recall that unit vectors are special vectors thathave a magnitude of one.

To create a unit vector in n-space, we use the same formula as 2-space:

u =

u = 1

|u| Using +1 gives a unit vector inthe same direction as u andusing -1 gives a unit vector in

the opposite direction.

(u1, u2, , un)

(u1, u2, , un)

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UNIT VECTORS

Ex. 14 Unit vectors-done in class

Ex. 15 Linear combinations-done in class

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CROSS PRODUCT

In 3-space it is often useful to create a vector thatis perpendicular to two given vectors. To do this, weuse the cross product.

i j k

u1 u2 u3

v1 v2 v3

If u = (u1, u2, u3) and v= (v1, v2, v3) are two vectors in 3-space, thenthe cross product is defined to be

u x v =

Note that this is a determinant that returns a

vector in i, j, k notation!

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CROSS PRODUCT

Because of the determinant, the cross product hassome very nice properties both algebraic andgeometric.

Thm. 16 Properties of the Cross Product-done in class

Proof-done in class

Thm. 17 Some Geometric Properties

-done in class

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VECTOR PROJECTIONS

In N-space, vector projections are calculated in thesame manner as in 2-space.

uv

|v|2vproj vu =

(uv)u -

|v|2vcomponent of u perpendicular to v =

uv

|v|component of u in the direction of v =

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VECTOR PROJECTIONS

Ex. 18 Creating a Projection in 3-space- done in class

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CAUCHY-SCHWARZ INEQUALITY

This inequality is a way of relating the dot productto a product of magnitudes. We need it to prove the

(more important) Triangle Inequality.

|uv| |u| |v|

Thm. 19 (Cauchy-Schwarz Inequality)

Let u and vbe vectors in n-space. Then

absolute value ofthe dot product

multiplication ofmagnitudes Cauchy-SchwarzNicole Scherzinger

= Nicole Cauchy-

Schwarzinger Inequality?

+

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CAUCHY-SCHWARZ INEQUALITY

This inequality is a way of relating the dot productto a product of magnitudes. We need it to prove the

(more important) Triangle Inequality.

uv|u| |v| cos()

Proof (Cauchy-Schwarz Inequality)

The proof comes from analyzing the cosine formula:uv

cos() =

|u| |v|

=

|uv||u| |v| |cos()| =

|uv||u| |v|

take absolute values

of both sides

|cos()| returnsvalues between zeroand one

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TRIANGLE INEQUALITY

What is the shortest path from A to B?Should we go straight from A to B? Or take a

detour through point C?

A

B

C

A straight line from A to B is always theshortest path (in n-space).

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TRIANGLE INEQUALITY

Lets write these paths with vectors:

A

B

C

u

v

u+vWe can usetheparallelogramlaw ofaddition toshow the

path from A toB is u+v.

E E

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TRIANGLE INEQUALITY

A

B

C

u

v

u+v

Thm. 20 (Triangle Inequality):Let u and vbe vectors in n-space. Then

|u + v| |u| + |v|

Proof (Triangle Inequality)

-done in class

E N F L NE ND PL NE

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EQUATIONS OF LINES AND PLANES

Consider a line L in 3-space that is parallel to thevector v= (a, b, c) and such that the pointP0(x0, y0, z0) lies on L. We will try to describe L interms of vand P0.

L

v

P0

O

EQUATION OF LINE AND PLANE

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Choose another point P(x, y, z) that lies on L. Createvectors r and r0 by connecting the origin to P and toP0, respectively.

L

v

P0 P

O

r0 r


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By reversing the vector r and using vector addition,we can create a vector on L that is parallel to v.This new vector has an initial point at P0 and aterminal point at P.

L

v

P0 P

O

r0 -r


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Fact: two vectors are parallel if they are scalarmultiples of one another. This lets us write the bluevector as a scalar multiple of v! *sneaky*

L

v

P0 P

O

r0 -r

-r + r0=tv

r = r0tv = (x0, y0, z0) t(a, b, c)


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This is called the vector equation of a line in3-space.

r = (x0, y0, z0) t(a, b, c)

If we use r = (x, y, z), we can compare each component to get theparametric equations of a line.

(x, y, z) = (x0, y0, z0) t(a, b, c)

x = x0 tay = y0 tbz = z0 tc


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Finally, we can solve for t in each parametricequation to get what are called the symmetric

equations of a line.

= tx x0

a

= ty y0

b

= tz z0

c


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Finally, we can solve for t in each parametricequation to get what are called the symmetric

equations of a line.

x x0

ay y0

b= z z

0

c=


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To get at an equation for a plane , we need thedefinition of a normal vector.

Any vector that is perpendicular to all vectors in a given plane iscalled a normal vector and is denoted by n.

n


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If we fix the point P0(x0, y0, z0) on the plane andtake any other point P(x, y, z), we can create the

vector P0P.

Since P0P is a vector in the plane, if we use the dot product on P0Pand n = (a, b, c), we should get zero.

n

P0

P


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Doing some simplification to get to standard form:

n

P0

P

P0P n = (x x0)a + (y y0)b + (z z0)c = 0

ax + by + cz = d = ax0 + by0 + cz0


8/3/2019 Vectors - 3.1, 3.2, 3.3, 3.4, 3.5

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Ex. 21 Some Line and Plane Questions- done in class


PYTHAGORAS OF SAMOS

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PYTHAGORAS OF SAMOS(~570BC - ~495BC)

- created the famous Pythagorean school for studyof arithmetic, geometry, music and astronomy (thiswas known as quadrivium and all educated people

needed to know these things)- the schools philospohy rested on the whole numbers, and that wholenumbers could explain all of man and matter

- eventually it was found that root 2 was irrational; this went against

the philosophy of the school so it was kept a secret; Hippasusparished at sea for disclosing this information to the outside world

- travelled extensively as a young adult and

migrated to a Greek colony in southern Italy

- the Pythagorean school became too influential for the governmentof Italy to handle, so it was burned down; Pythagoras fled and was

d d d f 5

vectors - 3.1, 3.2, 3.3, 3.4, 3.5

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