Vector in R2 and R3

Review of Coordinate Systems

•Used to describe the position of a point in space

•Common coordinate systems are:

– Cartesian

– Polar

Cartesian Coordinate System

•Also called rectangular coordinate system

•x- and y- axes intersect at the origin

•Points are labeled (x,y)

Polar Coordinate System

•Origin and reference line are noted

•Point is distance r from the origin in the direction of angle , from reference line

– The reference line is often the x-axis.

•Points are labeled (r,)

Polar to Cartesian Coordinates

•Based on forming a right triangle from r and

•x = r cos

•y = r sin

•If the Cartesian coordinates are known:

2 2

tany

x

r x y

Vectors in Two and Three

Dimensions

Algebraic and geometric

points of view

Algebraic Definitions

• A vector in R2 is an ordered pair of real numbers, e.g., (1,–3) or (π,√2). In general we write a = (a1, a2) for a vector in R2.

• A vector in R3 is an ordered triple of real numbers, e.g., (7,18,–9) or (e,0,–2). In general we write a = (a1, a2, a3) for a vector in R3.

Arithmetic Operation:

Vector Addition

• Add vectors in the same space by adding component-

wise.

• Examples

– (1, –3) + (5, 7) = (6, 4).

– (√2, 0, 1) + (–2, 5, –1) = (√2 – 2, 5, 0).

– (4, –1, 8) + (1, 2) is not defined.

• In general, if a = (a1, a2, a3) and b = (b1, b2, a3), then

a + b = (a1+b1, a2+b2, a3+b3). (Similar for vectors in R2.)

Arithmetic Operation:

Scalar Multiplication

• Multiply a vector by a single real number (called a

scalar) by multiplying the components of the

vector by that real number.

• Examples

3(1, 7) = (3, 21).

–2(1, 0, –3) = (–2, 0, 6).

• In general, if a = (a1, a2, a3) and k R, then

ka = (ka1, ka2, ka3). (Similar for vectors in R2.)

Unit vectors

• Length / magnitude = 1

• In R2

j

i

• In R3

y

z

x

j

k

i

y

x

• a = a1i + a2j

• b = =b1i + b2j + b3j

Length of vectors

• ||a|| =

• ||b|| =

2

2

2

1 aa

2

3

2

2

2

1 bbb

Some Properties

• For all a, b, c Rn (n = 2 or 3) and k, l R, we have:

1. a + b = b + a.

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

3. There’s a special vector 0 such that a + 0 = a.

4. k(a + b) = ka + kb.

5. (k + l)a = ka + la.

6. k(la) = (kl)a = l(ka).

Geometric Ideas

a

a

A vector a Rn (n = 2 or 3) can be represented by an

arrow from the origin to the point (a1, a2) or (a1, a2, a3).

(a1, a2)

(a1, a2, a3)

y

x

R2

y

z

x

R3

However, any parallel translate of the arrow from

the origin may also serve as a representation of a.

y

x

R2

y

z

x

R3

Any one of the arrows shown above represents the

vector a. The red arrow is referred to as the

position vector of the point (a1, a2) or (a1, a2, a3).

a a

Visualization of Vector Addition

b

a

Translate b so its tail is

at the head of a, then add

arrows tail-to-head.

Or, a + b is the diagonal

of the parallelogram

determined by a and b.

b (translated) a + b

Adding Some Vectors Graphically

•When you have many vectors, just keep repeating the process until all are included.

•The resultant is still drawn from the tail of the first vector to the head of the last vector.

u

v

w

s

Adding Vectors, Rules

•When two vectors are added, the sum is independent of the order of the addition.

– This is the Commutative Law of Addition.

• u + v = v + u

u

u

v

v

Adding Vectors, Rules cont.

•When adding three or more vectors, their sum is independent of the way in which the individual vectors are grouped.

– This is called the Associative Property of Addition.

• u + (v + w) = (u + v) + w

u

v

w

u

v

w

Visualization of

Scalar Multiplication

a

Stretch or compress a by the scalar |k|.

Reverse the sense of the arrow if k < 0.

2a

(1/3)a

–a

Subtraction of Vectors

b

a

Vector a – b is a + (–b).

a + b

a – b is the other

diagonal of the

parallelogram

determined by a

and b.

–b

a – b

a – b

–b (translated)

Displacement Vector

Between Two Points

P2

y

z

x

P1

P1P2

O

The displacement vector from the point P1(a1, a2, a3)

to the point P2(b1, b2, b3), denoted P1P2, is obtained by

subtracting position vectors.

P1P2 = OP2 – OP1

= (b1, b2, b3) – (a1, a2, a3)

= (b1– a1, b2 – a2, b3 – a3).

The Dot Product

A scalar product

Definition

a

a b = ||a|| ||b|| cos ,

where , for 0 ≤ ≤ π, is the angle between

a and b.

b

Theorem

• Let a = (a1, a2, a3) and b = (b1, b2, b3) and be vectors in R3.

• Then the dot product a b is

a b = a1b1 + a2b2 + a3b3.

(There’s an analogous definition for vectors in R2.)

Example (1, –3, 2) (2, 1, 5) = 12 + (–3)1 + 25 = 9.

2121 bbaa wv

1542 wv

8 5 3

jiwjiv 2211 and where baba

1

This is called the dot product. Notice the answer is just a number NOT a

vector.

find ,4 and 52 If wvjiwjiv

Angles between Vectors

a

b

When a and b are nonzero vectors, we can find the angle between them by

Hence a is perpendicular (orthogonal) to b if and only if a b = 0.

cos1 a b

a b

.

If and are two nonzero vectors, the angle

, 0 < , between and is determined

by the formula

u v

u v

vu

vu

cos

The dot product is useful for several things. One of the important uses

is in a formula for finding the angle between two vectors that have the

same initial point.

u

v Technically there are two angles between these vectors,

one going the "shortest" way and one going around the

other way. We are talking about the smaller of the two.

.3+4= and 2=between angle theFind jivjiu

v u

vu cos 5383142 vu

u 2 1 52 2

v 4 3 16 9 25 52 2

5

1

55

5cos

v u

vu

v i j 4 3

jiu 2 4.63

5

1cos 1

Find the angle between the vectors v = 3i + 2j and w = 6i + 4j

The vectors have the same direction. We

say they are parallel because remember

vectors can be moved around as long as

you don't change magnitude or direction.

w v

wv cos

5213

818

676

26 1

01cos 1What does it mean when the angle

between the vectors is 0?

jiv 23

jiw 46

0 wv

Determine whether the vectors v = 4i - j and w = 2i + 8j are orthogonal.

08124 wv

The vectors v and w are orthogonal.

If the angle between 2 vectors is , what would their dot product be?

2

v u

vu cos

Since cos is 0,

the dot product must be 0.

2

2

Vectors u and v in this case are called orthogonal.

(similar to perpendicular but refers to vectors).

compute their dot product

and see if it is 0

w = 2i + 8j

v = 4i - j

Compute the dot product of each pair of vectors:

ii

jj

ji

Given the vectors u = 8i + 8j – 3k and v = —10i + 11j – 2k find the following.

vu

vv

uv

Given the vectors: u = 8i + 8j – 3k, v = —10i +11j – 2k, and w = 9i + 7j + 5j ; find the following.

wvu

wwuv 123

wuvu

Find the angle θ in degrees measured between the vectors u = 10i + 3j and v = 1i — 7j .

Determine if the pair of vectors is orthogonal, parallel, or neither.

jivjiu3

162and38

jivjiu 921and37

Algebraic Properties

1. a a ≥ 0; a a = 0 if and only if a = 0.

2. a b = b a.

3. (ka) b = k(a b) = a (kb).

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

Connection to Length

a a12 a

22 a

32 a a

y

z

x a2

a3

a1

a

Normalization of a Vector

If a ≠ 0, a unit vector u in the direction of a is given by

||u|| = 1

u a

a.

Projections

• The next slide shows representations the same initial point P.

• If S is the foot of the perpendicular from R to the line containing with representation projection of b onto a and is denoted by proja b.

and of two vectors and withPQ PR a b

, then the vectorPQ is called the PS vector

Projections (cont’d)

Projections (cont’d)

• The scalar projection of b onto a is the length |b|cos θ of the vector projection.

• This is denoted by compa b, and can also be computed by taking the dot product of b with the unit vector in the direction of a.

Vector Projection

Assume a ≠ 0.

proja b a b

a a

a

projab

b

a

The Cross Product

A vector product

Definition

a b a2 a3

b2 b3i a1 a3

b1 b3ja1 a2

b1 b2k

.

321

321

bbb

aaa

kji

bax

a = a1i + a2j + a3k and b = b1i + b2j + b3k

Coordinate Formula

a b =

(a2b3 – a3b2)i – (a1b3 – a3b1)j + (a1b2 – a2b1)k

Geometric Definition

• Let a = (a1, a2, a3) and b = (b1, b2, b3) be

vectors in R3.

• The cross product a b is the vector in

R3 with the following features:

The magnitude (length) of a b is the area of the parallelogram determined by a and b.

a

b

||a b|| = ||a|| ||b|| sin

The direction of a b (if nonzero) is perpendicular to both a and b and is such that (a, b, a b) is right-handed.

a

b

a b

Algebraic Properties

1. a b = – b a

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

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

4. (ka) b = k(a b) = a (kb)