lesson 2-vectors-and-the-dot-product-1202327813713073-4

Section 9.2–3Vectors and the Dot Product

Math 21a

February 6, 2008

Announcements

I The MQC is open: Sun-Thu 8:30pm–10:30pm, SC 222I Homework for Friday 2/8:

I Section 9.2: 4, 6, 26, 33, 34I Section 9.3: 10, 18, 24, 25, 34I Section 9.4: 1*

Outline

VectorsAlgebra of VectorsComponentsStandard basis vectorsLength

The Dot ProductWorkConceptPropertiesUses

What is a vector?

Definition

I A vector is something that has magnitude and direction

I We denote vectors by boldface (v) or little arrows (~v). One isgood for print, one for script

I Given two points A and B in flatland or spaceland, the vectorwhich starts at A and ends at B is called the displacement

vector−→AB.

I Two vectors are equal if they have the same magnitude anddirection (they need not overlap)

Vector or scalar?

DefinitionA scalar is another name for a real number.

Example

Which of these are vectors or scalars?

(i) Cost of a theater ticket

scalar

(ii) The current in a river

vector

(iii) The initial flight path from Boston to New York

vector

(iv) The population of the world

scalar

Vector or scalar?

DefinitionA scalar is another name for a real number.

Example

Which of these are vectors or scalars?

(i) Cost of a theater ticket scalar

(ii) The current in a river vector

(iii) The initial flight path from Boston to New York vector

(iv) The population of the world scalar

Vector addition

DefinitionIf u and v are vectors positioned so the initial point of v is theterminal point of u, the sum u + v is the vector whose initial pointis the initial point of u and whose terminal point is the terminalpoint of v.

vu + v

The triangle law

The parallelogram law

Opposite and difference

DefinitionGiven vectors u and v,

I the opposite of v is the vector −v that has the same lengthas v but points in the opposite direction

I the difference u− v is the sum u + (−v)

u− v

Scaling vectors

DefinitionIf c is a nonzero scalar and v is a vector, the scalar multiple cv isthe vector whose

I length is |c | times the length of v

I direction is the same as v if c > 0 and opposite v if c < 0

If c = 0, cv = 0.

−12v

Scaling vectors

If c = 0, cv = 0.

−12v

Scaling vectors

If c = 0, cv = 0.

−12v

Properties

TheoremGiven vectors a, b, and c and scalars c and d, we have

1. a + b = b + a

2. a + (b + c) = (a + b) + c

3. a + 0 = a

4. a + (−a) = 0

5. c(a + b) = ca + cb

6. (c + d)a = ca + da

7. (cd)a = c(da)

8. 1a = a

These can be verified geometrically.

Components defined

Definition

I Given a vector a, it’s often useful to move the tail to O andmeasure the coordinates of the head. These are called thecomponents of a, and we write them like this:

a = 〈a1, a2, a3〉

or just two components if the vector is the plane. Note theangle brackets!

I Given a point P in the plane or space, the position vector of

P is the vector−→OP.

FactGiven points A(x1, y1, z1) and B(x2, y2, z2) in space, the vector

−→AB

has components

−→AB = 〈x2 − x1, y2 − y1, z2 − z1〉

Components defined

Definition

a = 〈a1, a2, a3〉

−→AB

has components

−→AB = 〈x2 − x1, y2 − y1, z2 − z1〉

Components defined

Definition

a = 〈a1, a2, a3〉

−→AB

has components

−→AB = 〈x2 − x1, y2 − y1, z2 − z1〉

Vector algebra in components

TheoremIf a = 〈a1, a2, a3〉 and b = 〈b1, b2, b3〉, and c is a scalar, then

I a + b = 〈a1 + b1, a2 + b2, a3 + b3〉I a− b = 〈a1 − b1, a2 − b2, a3 − b3〉I ca = 〈ca1, ca2, ca3〉

Useful vectors

DefinitionWe define the standard basis vectors i = 〈1, 0, 0〉, j = 〈0, 1, 0〉,k = 〈0, 0, 1〉. In script, they’re often written as ı̂, ̂, k̂.

FactAny vector a can be written as a linear combination of thestandard basis vectors

〈a1, a2, a3〉 = a1i + a2j + a3k.

Useful vectors

DefinitionWe define the standard basis vectors i = 〈1, 0, 0〉, j = 〈0, 1, 0〉,k = 〈0, 0, 1〉. In script, they’re often written as ı̂, ̂, k̂.

FactAny vector a can be written as a linear combination of thestandard basis vectors

〈a1, a2, a3〉 = a1i + a2j + a3k.

Length

DefinitionGiven a vector v, its length is the distance between its initial andterminal points.

FactThe length of a vector is the square root of the sum of the squaresof its components:

|〈a1, a2, a3〉| =√

a21 + a2

2 + a23

Length

DefinitionGiven a vector v, its length is the distance between its initial andterminal points.

FactThe length of a vector is the square root of the sum of the squaresof its components:

|〈a1, a2, a3〉| =√

a21 + a2

2 + a23

Early vector users

I Caspar Wessel (Norwegian and Danish, 1745–1818)

I Jean Robert Argand (French 1768–1822),

I Carl Friedrich Gauss (German, 1777–1855)

I Sir William Rowan Hamilton (Irish, 1805–1865)

Outline

VectorsAlgebra of VectorsComponentsStandard basis vectorsLength

The Dot ProductWorkConceptPropertiesUses

DefinitionWork is the energy needed to move an object by a force.

If the force is expressed as a vector F and the displacement avector D, the work is

W = |F| |D| cos θ

where θ is the angle between the vectors.

Work is |F| times this distance

DefinitionWork is the energy needed to move an object by a force.

If the force is expressed as a vector F and the displacement avector D, the work is

W = |F| |D| cos θ

where θ is the angle between the vectors.

Work is |F| times this distance

DefinitionIf a and b are any two vectors in the plane or in space, the dotproduct (or scalar product) between them is the quantity

a · b = |a| |b| cos θ,

where θ is the angle between them.

Another way to say this is that a · b is |b| times the length of theprojection of a onto b.

ba · b is |b| times this length

DefinitionIf a and b are any two vectors in the plane or in space, the dotproduct (or scalar product) between them is the quantity

a · b = |a| |b| cos θ,

where θ is the angle between them.Another way to say this is that a · b is |b| times the length of theprojection of a onto b.

ba · b is |b| times this length

Geometric properties of the dot product

I Two vectors are perpendicular or orthogonal if their dot

product is zero (i.e., cos θ = 90◦ =π

I The law of cosines can be expressed as

|a + b|2 = |a|2 + |b|2 − 2 |a| |b| cos θ

= |a|2 + |b|2 − 2a · b

I In components, if a = 〈a1, a2, a3〉 and b = 〈b1, b2, b3〉, then

a · b = a1b1 + a2b2 + a3b3

|a + b|2 = |a|2 + |b|2 − 2 |a| |b| cos θ

= |a|2 + |b|2 − 2a · b

a · b = a1b1 + a2b2 + a3b3

|a + b|2 = |a|2 + |b|2 − 2 |a| |b| cos θ

= |a|2 + |b|2 − 2a · b

a · b = a1b1 + a2b2 + a3b3

More geometric properties of the dot product

FactThe angle between two nonzero vectors a and b is given by

cos θ =a · b|a| |b|

where θ is taken to be between 0 and π.

FactThe angle between two nonzero vectors a and b is

I acute if a · b > 0

I obtuse if a · b < 0

I right if a · b = 0;

The vectors are parallel if a · b = ± |a| |b|.

I b is a positive multiple of a if a · b = |a| |b|I b is a negative multiple of a if a · b = − |a| |b|

The vectors are parallel if a · b = ± |a| |b|.I b is a positive multiple of a if a · b = |a| |b|

I b is a negative multiple of a if a · b = − |a| |b|

The vectors are parallel if a · b = ± |a| |b|.I b is a positive multiple of a if a · b = |a| |b|I b is a negative multiple of a if a · b = − |a| |b|

Examples

Example

Find the sum of the following pairs of vectors geometrically andalgebraically.

(i) a = 〈3,−1〉 and b = 〈−2, 4〉(ii) a = 〈0, 1, 2〉 and b = 〈0, 0,−3〉

What is the angle between the two vectors in each case?

Solution

(i) a + b = 〈1, 3〉, |a| =√

10, |b| =√

20. So

=−6− 4√

−10√10√

20= − 1√

2=⇒ θ =

(ii) a + b = 〈0, 1,−1〉, while

cos θ =0 + 0− 6√

9= − 2√

Examples

Example

Find the sum of the following pairs of vectors geometrically andalgebraically.

(i) a = 〈3,−1〉 and b = 〈−2, 4〉(ii) a = 〈0, 1, 2〉 and b = 〈0, 0,−3〉

What is the angle between the two vectors in each case?

Solution

(i) a + b = 〈1, 3〉, |a| =√

10, |b| =√

20. So

=−6− 4√

−10√10√

20= − 1√

2=⇒ θ =

(ii) a + b = 〈0, 1,−1〉, while

cos θ =0 + 0− 6√

9= − 2√

Properties

FactIf a, b and c are vectors are c is a scalar, then

1. a · a = |a|22. a · b = b · a3. a · (b + c) = a · b + a · c

4. (ca) ·b = c(a ·b) = a · (cb)

5. 0 · a = 0

Example

The dot product can be used to measure how similar two vectorsare. Consider it a compatibility index. If two vectors point inapproximately the same direction, we get a positive dot product. Iftwo vectors are orthogonal, we get a zero dot product. If twovectors point in approximately opposite directions, we get anegative dot product.Consider the following categories,

1. Football

2. Sushi

3. Classical music

Now create a vector in R3 rating your preference in each categoryfrom −5 to 5, where −5 expresses extreme dislike and 5 expressesadoration. Dot your vector with your neighbor’s.

Example

Fifi, a poodle, drags her owner along a sidewalk that is 200 meterslong. If Fifi exerts a force of two newtons on the leash, and theleash is at an angle 45◦ from the ground, how much work does Fifido?

lesson 2-vectors-and-the-dot-product-1202327813713073-4

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