CSEC Maths - Sample Chapter

9 CHAPTER

Vectors, matrices and transformations

Objectives By the end of the chapter you should be able to: • explain concepts associated with vectors: concept of a vector, magnitude, unit vector, direction, parallel vectors, equal vectors, inverse vector as well as scalars: scalar multiples, magnitude, etc • simplify expressions involving vectors; work with vector algebra: addition, subtraction, scalar multiplication and vector geometry, triangle law, parallelogram law  a  • write down the position vector of a point P(a, b) as OP =   where O is the b   origin (0, 0) • describe displacement and position vectors, including the use of coordinates in the x–y plane, to identify and determine displacement and position vectors • determine the magnitude of a vector, including unit vectors (magnitude 1) • determine the direction of a vector • use vectors to solve problems in geometry and to explain the significance of points in a straight line, parallel lines, displacement velocity, weight • explain basic concepts associated with matrices: concept of a matrix, types of matrix, row, column, square, rectangular, order of a matrix • solve problems involving matrix operations such as addition and subtraction of matrices of the same order, scalar multiples, multiplication of conformable matrices, equality and non-­commutativity of matrix multiplication • find the determinant of a 2 × 2 matrix, define the multiplicative inverse of a non-singular square matrix, work with identity square matrices • find the inverse of a non-singular 2 × 2 matrix, the determinant and adjoint of a matrix • determine a 2 × 2 matrix associated with a specified transformation, for ­example: • reflection in the x-axis, y-axis, the lines y = x and y = –x • rotation in a clockwise or anticlockwise direction about the origin (the general rotation matrix) • enlargement with centre at the origin • other important transformations • use matrices to solve simple problems in arithmetic, algebra and geometry • discuss the use of matrices to present data, the equality of matrices, the use of matrices to solve linear simultaneous equations with two unknowns.

(Problems involving determinants are restricted to 2 × 2 matrices.)

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You should already know: • Number theory and computation

• Measurement

• Algebra

• Fractions and decimals

• Geometry and trigonometry

• Relations and functions

Introduction These instructions were given to someone trying to locate a building: ‘Walk straight ahead for two blocks, turn right and walk for another block then walk three more blocks in a southern direction.’ The instructions include direction as well as distance or magnitude. This is what vectors are all about. There are many other everyday examples of vector applications in real life, ranging from the simple to the complex.

Vectors are used, as you have just seen, to locate objects and individuals. They are also used to describe the behaviour of people and objects subjected to forces. In sports, the force and direction with which the javelin, football, other missiles, even including the human body (in high jump and long jump) are launched, are crucial in acquiring the desired results or success, for example, the greatest distance a javelin can be thrown is achieved when it is launched at 45° to the horizontal.

In the more professional and technical domain, vectors are essential in engineering, whether in building a bridge sturdily and safely to withstand forces in various directions, designing a vehicle or other machines, navigation – ensuring that wind speed and direction are adequate for aeroplanes, boats and other crafts. One of the expressions used by air traffic control personnel on the approach of a plane is, ‘Expect vectors for the visual approach.’

In this chapter you will learn how to use vectors and matrices to model and solve real-world problems. Additionally, you will learn to define and describe various linear transformations, their application in everyday problem-solving situations and how they can be combined with vectors and matrices to simplify the solution of these everyday problems further.

Examples of vectors and matrices in real life

Aircraft are given instructions by the control tower involving direction and distance as they land.

Distances between points on a graph have direction and distance.

Communications between countries involve messages moving various distances in many directions.

The instrument panels of a helicopter give the pilots vital information, especially when they are flying over airfields at night.

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9.1 Concepts associated with vectors In previous work the quantities that you have worked with have, in the main, been scalar quantities, which have magnitude but no direction. However, you will be aware that some quantities, such as forces, act in particular directions. Weight, for example, acts downwards while velocity acts in the direction of motion.

Quantities that have direction as well as magnitude are vectors.

Key terms A scalar quantity is a quantity that has magnitude only, without direction.

The magnitude of a quantity is its size.

A vector quantity is a quantity with both magnitude and direction; for example, a force of 20 N acting horizontally or a velocity of 10 m/s acting south.

9.1.1 Vector representation

Example 1

A vector may be represented by a line segment of given magnitude or length, with an arrow at one end indicating the direction of action of the vector. The line segment may be drawn to scale.

Represent the vector 20 N due west.

The vector is represented by a horizontal line, drawn to a scale of 1 cm : 2 N, with an arrow pointing west.

20 N

A vector starting at P and ending at Q can be represented with capital letters and an arrow as PQ. Vectors are sometimes represented by a lower-case or common bold letter, for example, PQ = a. The magnitude of vector PQ may be expressed as |PQ| and the magnitude of a may be expresse as |a| or a.

9.1.2 Scalar multiplication You know that a scalar quantity has size, or magnitude, but no direction. Scalar multiplication increases or decreases the magnitude of a vector by a factor.

420

Example 2

A vector is given as PQ = 5 m/s2 due south. Evaluate 5PQ.

Example 3

In general, if k is a real number, the vector kLM is also a vector in the direction of LM and of magnitude k times the magnitude of LM.

M is the midpoint of PR and MR = b. Write the vector 2PR in terms of b.

5PQ = 5 × 5 = 25 m/s2 due south

2PR = 4 × 2MR

=4×b

= 4b

9 Vectors, matrices and transformations

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12

From a position P on the playing field a girl runs 20 m in a north-easterly direction, then 15 m east, 24 m south, 12 m west and then returns to her starting point. a What was the total distance covered? b In what direction was her final run back to her starting point?

13

A vector LM has a magnitude of 2a units. R is the midpoint of LM so LR = RM = a.

Another vector LP is twice LM. Determine the resultant of this combination of the vectors, in terms of a.

111

33 11 2LM 22LM −− RM RM −− ++3RL 33RL −− 11 PL LM— RM— RL— PL ++ 22 RP RP 22 44 22

14

Determine the magnitude of resultant of these three vectors, in terms of a, b and c.

a b 45° c

15

A

In the diagram, AD represents vector a, BC represents vector b and DC represents vector c. P is the midpoint of AD and Q is the midpoint of BC.

c AB

d DQ

e BP

f PC

a

D

c

Write, in terms of a, b and c, the vectors: a AC b BD

P

B

b C

Q

g PQ.

9.2 Position vectors 9.2.1 Concept and description of position vectors and unit vectors The position of any point may be described in terms of any other fixed point. Graphically, the position vector of point P from the origin O is drawn as an arrow from O to P. Mathematically it is expressed in terms of unit vectors. In diagrams, lower-case or common letters such as a, b, p, q may also be used to denote the magnitude and direction of a position vector.   A position vector can be two-dimensional  x  or y   x three-dimensional  y . z  

Note

Using column vectors to describe vectors in this way is explained in Section 9.2.4.

Key term A unit vector is a vector with a magnitude of 1. It is sometimes called a direction vector.

Key term A position vector, sometimes called a location vector, is a vector that represents the position of a point P in space in relation to a fixed reference point such as the origin (O).

9.2 Position vectors

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Exercise 9B 1

Identify the position vectors in the diagram, representing them as: a unit vectors

2

3

4

Represent the position vectors a = 4i + 3j and b = diagram.

Q

10

   2  on one 6  

9 8 7 6

 −3  Draw a diagram to represent the position vectors OA OA =   and 8    2  OB OB =   .  −9    Determine the magnitude and direction of these position vectors. a OR = 3i + 4j

5

b column vectors.

y

4 3 2 1 0

 6  b OM OM =   8  

Using vector notation:

P

5

0

1

2

3

x

4

y

a name the position vector shown in the diagram in two ways and determine its magnitude and direction

4

b write two equations representing the position vector.

2

P

3

a

1 0

0

2

4

6

x

8 10 12

6

Write the unit vector that acts in the same direction as position vector 8i – 12j.

7

The diagram shows two position vectors, OP = a and OQ = b.

y P

Express 2PQ + QP in terms of a and b.

a

Q

b O

8

The vectors in the diagram (not drawn to scale) are parallel to each other.

MN = 3a, PQ = 2a, RS = 5a, UV = a

R

a Express their sum in terms of a.

U

x

M

N

P

Q S V

b Express 2MN – PQ + RS – 8UV in terms of a.

9

The diagram shows two position vectors, OW and OS. a Represent each position vector as a column vector. b Determine the sum of the position vectors, expressing your answer as a column vector. c Express OS – OW as a column vector.

y

W

6 5 4 3

S

2 1 0

0

1

2

3

4

5

6

7

8

9 10

9.2 Position vectors

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The order of the resultant matrix is:

Example 20

Note

(number of rows in matrix 1) × (number of columns in matrix 2).

The two outer values of the orders of the matrices, when they are written as a product, give the order of the resultant while the two inner values satisfy the condition for multiplication if they are equal.

  Investigate the product ( 4 6 1)  5  . 7    The order of the first matrix is 1 × 3.

The order of the second matrix is 2 × 1.

This multiplication is not possible since the number of columns in the first matrix is not equal to the number of rows in the second.

Writing the product in terms of the orders of the matrices, (1 × 3)(2 × 1), shows that the two inner numbers are not equal so the necessary condition for multiplication is not satisfied.

9.3.4 Matrix notation, commutativity and equality Matrix notation and commutativity Matrices are usually denoted by capital letters,     2 3 3 1 for example, A =  ,B =  .  4 5  1 6    

Key fact In summary, you can only multiply a matrix of order m × n by a matrix of order p × q if n = p.

Example 21

Generally, multiplication of two matrices is not commutative: A × B ≠ B × A.     2 3  and 3 1 A= B=  4 5  1 6     Show that A × B ≠ B × A. Using the rules of matrix multiplication:    2 3   3 1 AB =   4 5  1 6       6 + 3 2 + 18  =  12 + 5 4 + 30      9 20  =  17 344   

   3 1  2 3  BA =   1 6   4 5      6+4 9+5 =  2 + 24 3 + 30      10 14  =  26 333   

So A × B ≠ B × A.

Equal matrices Two matrices are equal if they have the same order and the elements in corresponding positions are the same.

436

Key term Equal matrices have the same order and the elements that are in corresponding positions are identical.

9 Vectors, matrices and transformations

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End of chapter summary Concepts associated with vectors • A scalar quantity is a quantity that has magnitude only, without direction. A vector quantity is a quantity with both magnitude and direction; for example, a force of 20 N acting horizontally or a velocity of 10 m/s acting south.

• The magnitude of a quantity is its size.

• Scalar multiplication increases or decreases the magnitude of a vector by a factor.

• Two vectors are parallel if they act in the same direction, for example, AB || LM. A

B L

M

• Two vectors are equal if they are parallel and have the same magnitude, for example, Q

P R

10 N

S

• Two vectors are inverse if they have the same magnitude but opposite directions, for example, HK and SR are inverse vectors. Generally the inverse of any vector AB is –AB = BA. H

K

R

S

• Vector algebra involves the addition and subtraction of vectors and multiplication of a vector by a scalar.

• The general rule for adding two or more vectors is that the vectors must be aligned head to tail while maintaining their original direction. The combined vector is the resultant, often represented by r.

• Subtracting one vector from another is equivalent to adding the inverse of the subtracted vector to the first vector.

• The resultant of two or more vectors is the vector that 'results' from the effects of the individual vectors.

• The direction of the resultant AC is given by the angle q between the resultant and the northerly direction, AB. • The magnitude of AC is evaluated by applying Pythagoras’ theorem to the right-angled triangle.

• The triangle law states that if two adjacent sides of a triangle, drawn in order, represent the magnitudes and directions of two vectors acting from the same point, then the third side of the triangle represents the resultant of the two vectors in magnitude and direction.

• The parallelogram law states that if two vectors acting at a point are represented in magnitude and direction by two adjacent sides of a parallelogram, drawn from the same point, then the resultant of the two vectors is represented in magnitude and direction by the diagonal of the parallelogram, drawn from the same point.

Position vectors • A unit vector is a vector with a magnitude of 1. It is sometimes called a direction vector.

452

• A position vector, sometimes called a location vector, is a vector that represents the position of a point P in space in relation to a fixed reference point called the origin (O) and is denoted by an arrow from the origin x   x • A position vector can be two-dimensional   or three-dimensional  y . y z     9 Vectors, matrices and transformations

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Examination-type questions for Chapter 9 2 3  0 5 1 Matrices A and B are given as A =  and B =   .  1 4 7 6 a Show by matrix multiplication that AB ≠ BA. b Determine the inverse of B. c Determine the product BB–1. d Use a matrix method to solve the simultaneous equations: 3x – 2y = 4 x–y=1  5   8  2 Two position vectors are given as OP OP =   and OF OF =   .  10   16      1 A point R is on OP such that OP = 5OR and a point L is on OF such that OL is OF. 4 a Draw a sketch of each position vector, showing points R and L.   b Write OR and OL as column vectors in the form  x . y   c Determine the vector sum of PF and RL.   a 1 maps (2, 3) onto (5, 8).

3 The matrix W =  a b   a Find the values of a and b. b Determine the image of (6, –5) under the transformation W. 4 The graph shows the points A, B and C in relation to the origin O.   a Write the position vectors OA, OB and OC in the form  x . y   b Determine the magnitude and direction of the

y 10 8 6

position vector OB. c Write the vector BC as a column vector.

2 0

PQ = t and SR = 2t. QR = v.

State two parallel vectors.

Write PQ + SR in terms of t.

Determine SQ in terms of t and v.

Write in terms of t and v the vector PS. Write in terms of t and v the vector PR. Write in terms of t and v the resultant of PQ – SR.

A

4

5 PQRS is a trapezium with sides PQ and SR parallel. a b c d e f

B

12

P

C

O 0

2

4

6

8 10 12 14 16

Q

S

Examination-type questions

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R

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10 CHAPTER

The School-Based Assessment in Mathematics

Introduction The School-Based Assessment (SBA) is a practical, effective form of assessment used by many learning institutions. It has been part of the assessment used by the Caribbean Examination Council (CXC®) since it was first introduced in 1979.

SBAs help to monitor your performance over a period of time. They are meaningful assessments that help to create greater motivation. You actively participate in your own learning as well as in your own assessment. Through the SBA project activities, you are presented with various ways of demonstrating your knowledge, skills and attitudes.

One of the main reasons for introducing the SBA for Mathematics in 2018 was to help apply the mathematics you do in schools to everyday or real-world mathematics. Through the collection, representation and analysis of data the theoretical side of mathematics is now linked to real-world applications.

In the process, you will practise various problem-solving strategies such as: • learning to define the problem in a personal way • clarifying the problem through discussion with teachers, family, peers and resource persons in the community • developing your own strategies, using words, charts, tables, symbols, models, algorithms, diagrams and figures • evaluating the results • developing a positive attitude towards mathematics and its use • extending the mathematical processes to the understanding of other subjects in the curriculum.

The SBA project activities are linked to the syllabus and the marks obtained through these activities form part of your course assessment, which is then included in your final grade. It is therefore important that you score as highly in your SBA project as you can. You and your teacher can decide together on the project that you want to work on.

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Managing the project Your Mathematics School-Based Assessment (SBA) consists of one project, which has to be marked by the teacher, according to CXC® guidelines.

The use of projects is an ideal way of applying mathematical concepts, skills and procedures and to investigate and solve everyday mathematical problems, such as planning an outing, shopping, preparing a meal, building a simple structure or competing in a sporting event.

The role of the teacher is to: • provide guidance to students in terms of choosing their project topics • provide guidance and assistance throughout the SBA, ensuring that project content is also in line with the syllabus • help maintain the continuity of the projects within scheduled classes and outside class time • assess the projects, using the CXC® mark scheme provided and, together with students, keep both soft and hard copies.

Possible project activities include: • an explanation of mathematical ideas • carrying out practical tasks, using mathematical instruments such as rulers, compasses and protractors • drawing, constructing, measuring, looking for patterns, counting, weighing, etc.

• the use of technological devices such as calculators and computers • performing calculations mentally, with paper and pencil, using calculators, etc.

• oral responses to mathematics questions asked by teachers, peers and other interested parties • identification of sections of the project to be done within normal class activities (supervised by the teacher) and outside normal class (in students’ own time).

Specific activities for students • State the task (nature, scope and focus).

• Devise a plan (what to do, how to do it), identifying materials needed, procedures to follow.

• Carry out the plan, procedure and/or activities.

• Record activities, stating what was done, how it was done, why it was done, using mathematical words and phrases, diagrams, tables, figures or charts.

• Reach a conclusion, including describing findings, comments, suggestions for improvement of findings, putting findings to greater use.

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