Chapter 12Vectors

A. Vectors and scalarsB. Geometric operations with vectorsC. Vectors in the planeD. The magnitude of a vectorE. Operations with plane vectorsF. The vector between two pointsG. Vectors in spaceH. Operations with vectors in spaceI. ParallelismJ. The scalar product of two vectors

Opening Problem

An airplane in calm conditions is flying at 800 km/hr due east. A cold wind suddenly blows from the south-west at 35 km/hr, pushing the airplane slightly off course.

Things to think about:a. How can we illustrate the plane’s movement and the wind using a scale diagram?b. What operation do we need to perform to find the effect of the wind on the airplane?c. Can you use a scale diagram to determine the resulting speed and direction of the airplane?

Vectors and Scalars

Quantities which have only magnitude are called scalars.

Quantities which have both magnitude and direction are called vectors.

The speed of the plane is a scalar. It describes its size or strength.

The velocity of the plane is a vector. It includes both its speed and also its direction.

Other examples of vector quantities are:

acceleration force

Displacement momentum

Directed Line Segment Representation

We can represent a vector quantity using a directed line segment or arrow.

The length of the arrow represents the size or magnitude of the quantity, and the arrowhead shows its direction.

For example, farmer Giles needs to remove a fence post. He starts by pushing on the post sideways to loosen the ground.Giles has a choice of how hard to push the post and in which direction. The force he applies is therefore a vector.

If farmer Giles pushes the post with a force of 50 Newtons (N) to the north-east, we can draw a scale diagram of the force relative to the north line.

45o

N50N

Scale: 1 cm represents 25 N

Vector Notation

Geometric vector equality

Two vectors are equal if they have the same magnitude and direction.

Equal vectors are parallel and in the same direction, and are equal in length. The arrows that represent them are translations of one another.

Geometric negative vectors

AB and BA have the same length, but theyhave opposite directions.

A A

B B

Geometric operations with vectors

A typical problem could be:A runner runs east for 4 km and then south for 2 km. How far is she from her starting point and in what

direction?4 km

2 kmx km

N

S

W E

Geometric vector addition

Suppose we have three towns P, Q, and R.A trip from P to Q followed by a trip from Q to R has the same origin and destination as a trip from P to R.

This can be expressed in vector form as the sum PQ + QR = PR.

P

R

Q

“head-to-tail” method of vector addition

To construct a + b:

Step 1: Draw a.

Step 2: At the arrowhead end of a, draw b.

Step 3: Join the beginning of a to the arrowhead end of b. This is vector a + b.

Given a and b asshown, constructa + b.

ab

ab

a

b

a + b

THE ZERO VECTOR

The zero vector 0 is a vector of length 0.

For any vector a: a + 0 = 0 + a = aa +(-a) =(-a) + a = 0.

Find a single vector which is equal to:a. BC + CAb. BA + AE + ECc. AB + BC + CAd. AB + BC + CD + DE

A

E D

C

B

A

E D

C

B

BC + CA = BA

A

E D

C

B

BA + AE + EC = BC

AB + BC + CA = AA = 0

AB + BC + CD + DE = AE

Geometric vector subtraction

To subtract one vector from another, we simply add its negative.

a – b = a +(-b)

a

b

a

b

-b

a - b

For r, s, and t shown, find geometrically:

a. r – s

b. s – t – r

r

s t

r – s

r s

-sr – s

s – t – r

r

s

t

-ts – t – r

For points A, B, C, and D, simplify the following vector expressions:

a. AB – CB

b. AC – BC – DB

a. Since –CB = BC, then AB – CB = AB + BC = AC.

b. Same argument as part a.

AC – BC – DB = AC + CB + BD = AD

Vector Equation

Whenever we have vectors which form a closed polygon, we can write a vector equation which relates the variables.

Find, in terms of r, s, and t:

a. RS

b. SR

c. STO T

SR

r

t

s

a. Start at R, go to O by –r then go to S by s. Therefore RS = -r + s = s – r.

b. Start at S, go to O by –s then go to R by r. Therefore SR = -s + r = r – s.

c. Start at S, go to O by –s then go to T by t. Therefore ST = -s + t = t – s.

Geometric scalar multiplication

If a is a vector and k is a scalar, then ka is also a vector and we are performing scalar multiplication.

If k > 0, ka and a have the same direction.

If k < 0, ka and a have opposite directions.

If k = 0, ka = 0, the zero vector.

Given vectors r and s, construct geometrically:

a. 2r + s

b. r – 3s

r s

a. 2r + s

r s2r

s

2r + s

b. r – 3s

r

s

r

-3s

r – 3s

Vectors in the plane

In transformation geometry, translating a point a units in the x-direction and b units in the y-direction can be achieved using the translation vector

b

a

Base unit vector

All vectors in the plane can be described in terms of the base unit vectors i and j.

a. 7i + 3j

7i

3j

b. -6ic. 2i – 5jd. 6je. -6i + 3jf. -5i – 5j

The magnitude of a vector

v1

v2v

||.

||.

525

3

q

p

jiqp

b

a

findandIf

units

a

34

)5(3||

5

3.

22

p

p

units

q

b

29

)5(2||

5

252.

22

ji

Unit vectors

A unit vector is any vector which has a length of one unit.

are the base unit vectors in

the positive x and y-directions respectively.

1

0

0

1ji and

Find k given that is a unit vector.

k3

1

Knowing that is a unit vector,

then

k3

1

3

22

9

8

19

1

19

1

13

1

2

2

2

22

k

k

k

k

k

Operations with plane vectors

a

a1

a2

a1+b1

b

b1

b2a2+b2

a+b

.7

4

3

1baba

findandIf

Check your answer graphically.

4

5

73

41ba

Algebraic negative vectors

Algebraic vector subtraction

rqp

pq

rp

.

.

5

2

4

1

2

3

b

a

findandqandIf

6

2

24

31. pqa

1

4

542

213. rqpb

Algebraic scalar multiplication

qp

2qp

q

qp

32

1.

.

3.

3

2,

1

4

c

b

a

findFor

2

19

4

3312

1

2342

1

3

23

1

4

2

13

2

1.

5

8

321

224

3

22

1

42.

9

6

3

233.

qp

qp

q

c

b

a

If p = 3i – 5j and q =-i – 2j, find |p – 2q|.

p – 2q = 3i – 5j – 2(-i – 2j) = 3i – 5j + 2i + 4j = 5i – j

26)1(5|2| 22 qp

The vector between two points

O

A

B

a1 b1

a2

b2

Given points A(-1, 2), B(3, 4), C(4, -5) and O(0, 0), find the position vector of:a. B from O

b. B from A

c. A from C

7

5

52

4)1(.

2

4

24

13.

4

3

04

03.

CAc

ABb

OBa

[AB] is the diameter of a circle with center C(-1, 2). If B is (3, 1), find:

a. BC

b. the coordinates of A.

)3,5(

35

1241

1

4

2

1,

2

1

2

1),,(.

1

4

12

31.

isA

banda

banda

b

asoBCCABut

b

a

b

aCAthenbascoordinatehasAIfb

BCa

Vectors in space

3-D point plotter: demos #72 content disk

Recall the length of the diagonal of a rectangular prism (box),or if it’s easier, call it “3-D Pythagorean Theorem”

Illustrate the points:a. A(0, 2, 0)

b. B(3, 0, 2)

c. C(-1, 2, 3)

a. A(0, 2, 0)

b. B(3, 0, 2)

c. C(-1, 2, 3)

The vector between two points

If P is (-3, 1, 2), Q is (1, -1, 3), and O is (0, 0, 0), find:

a. OP

b. PQ

c. |PQ|

21124||

1

2

4

23

11

31

2

1

3

02

01

03

222

PQ

PQ

OP

Vector Equality

ABCD is a parallelogram. A is (-1, 2, 1), B is (2, 0,-1), and D is (3, 1, 4).

Find the coordinates of C.

A(-1,2,1)

B(2,0,-1)

D(3,1,4)

C(a,b,c)

)2,1,6(

,,

2

2

3

4

1

3

],[][

4

1

3

2

2

3

11

20

12

isCSo

candbaforsolve

c

b

a

ABDC

lengthsamethehavethey

DCtoparallelisABSince

c

b

a

DC

AB

Operations with vectors in space

Properties of vectors

Two useful rules are:

|2|,

2

3

1

aa findIf

Using the one of the properties of vectors, we know that

|ka| = |k| |a|Therefore

|2a| = 2|a|

units1422312||2 222 a

Find the coordinates of C and D:

)1,1,0(

1

1

0

1

3

1

2

3

5

2

2

2

1

3

1

32

52

21

.,

isC

ABOAOC

ABACandACOAOC

AB

ABfindtoneedweCofscoordinatethefindTo

)0,4,1(

0

4

1

1

3

1

3

3

5

2

3

3

isD

ABOAOC

ABADandADOAOD

Parallelism

Parallelism Properties

.2

3

2

13

42

12

2

121

3,21,2

3

21

2

,

rrand

ssthen

kkwithkforsolve

krandkks

s

k

r

kparallelareandSince baba

Unit vectors

If a = 3i - j find:

a. a unit vector in the direction of a

b. a vector of length 4 units in the direction of a

c. vectors of length 4 units which are parallel to a.

a. a unit vector in the direction of a

ji

aa

aa

10

1

10

3

10

110

3

1

3

10

1

||

1

1013||, 22

vectorunit

unitsvectoroflengththe

b. a vector of length 4 units in the direction of a

ji

ji

aa

10

4

10

12

310

14

||

14

c. vectors of length 4 units which are parallel to a.

10

4

10

12

10

4

10

12

10

4

10

12

||

144

||

144

||

1

iii

aa

aa

aa

and

parallelarewhichunitslength

unitslength

parallel

Find a vector b of length 7 in the opposite

direction to the vector

1

1

2

a

1

1

2

6

7

1

1

2

6

17

||

177

1

1

2

6

1

1

1

2

112

1

||

1222

b

b

aa

aa

negativeopposite

lengthofvectorunit

vectorunit

The scalar product of two vectors

There are two different types of product involving two vectors:

1. The scalar product of 2 vectors, which results in a scalar answer and has the notation v●w (read “v dot w”).

2. The vector product of 2 vectors, which results in a vector answer and has the notation vΧw (read “v cross w”).

Scalar product

ALGEBRAIC PROPERTIES OF THE SCALAR PRODUCT

qp

qp

qp

andbetweenangletheb

a

findandIf

.

.

:,

2

0

1

1

3

2