Introduction to Vectors Pg. 279 # 1 6, 8, 9, 10 OR WS 1.1 Sept. 7. Vector Addition Pg. 290 # 3, 4, 6, 7, OR WS 1.2 Sept. 8

Size: px
Start display at page:

Download "Introduction to Vectors Pg. 279 # 1 6, 8, 9, 10 OR WS 1.1 Sept. 7. Vector Addition Pg. 290 # 3, 4, 6, 7, OR WS 1.2 Sept. 8"

Transcription

1 UNIT 1 INTRODUCTION TO VECTORS Lesson TOPIC Suggested Work Sept Review of Pre-requisite Skills Pg. 273 # 1 9 OR WS 1.0 Fill in Info sheet and get permission sheet signed. Bring in $3 for lesson shells & $7 if you need a calculator Sept (1) 6.1 Introduction to Vectors Pg. 279 # 1 6, 8, 9, 10 OR WS 1.1 Sept (2) 6.2 Vector Addition Pg. 290 # 3, 4, 6, 7, OR WS 1.2 Sept (3) 6.3 Multiplication of a Vector by a Scalar Pg. 298 # 4, 5, 7a, 11, 13, 15, 17, 19 OR WS 1.3 Sept (4) 6.4 Properties of Vectors Pg. 306 # 3, 5, 7, 8ac, 9, 11 OR WS 1.4 Sept (5) 6.5 Vectors in R 2 and R 3 Pg. 316 # 3, 5, 6, 7, 9, 11, 13, 14, 15 OR WS 1.5 Sept (6) 6.6 Operations with Algebraic Vectors in R 2 Pg. 324 # 1, 3, 4, (5, 6)b, 8bd, 10, (13, 15)a Sept (7) 6.7 Vectors in R 3 Pg. 332 # 1, 2, 4, (5, 6)ac, 8, 9, 11, 12 Sept (8) 6.8 Linear Combinations and Spanning Sets Pg. 340 # 1, 3, 5, 7b, 9bd, 11, 12, 13b, 14, 15 Sept (9) Review for Unit 1 Test Pg. # 344 # 2a, 3 6, 7ab, 8, 11a, 12, 13, 15, 16, 18, 19, 21 Sept (10) TEST- UNIT 1

2 MCV 4U Lesson 1.1 Introduction to Vectors A vector is a mathematical quantity having both magnitude (size) and direction. Velocity is a vector - 60 km/h North is a vector quantity as is 40 Newtons down. A scalar is a mathematical quantity has magnitude or size only. NO DIRECTION. Speed is a scalar - 60 km/h is a scalar quantity as is 40 Newtons. A vector is represented as directed line segments with an arrow at one end to indicate the direction. The length of the line segment represents the magnitude of the vector. ALWAYS draw vector diagrams to scale. Draw angles as accurately as you can. The better the diagram, the more help it will be in solving the problem. If asked for a vector diagram DO NOT FORGET THE ARROWHEADS. (no arrows = no vector = no marks) When a single letter is used to denote a vector it is written in bold type a or it is written with an arrow over the a. It is faster to make the arrow if it only has a one sided point a ; it may also be written with a line over the a. I will use arrows with a single or double headed point. You may use whichever you wish except a. Sometimes we denote a vector using two capital letters when the vector extends from one point to another. In this case the first letter is always the starting or initial point of the vector and the second letter is the end or terminal point. B terminal point (tip) AB A Initial point (tail) The magnitude of a vector is denoted by placing absolute value signs around the vector. The magnitude of a is a. The magnitude of AB is AB. Vectors are equal iff their magnitudes and directions are the same. Opposite vectors have the same magnitude but opposite directions. The opposite of AB is AB or BA. When vectors have the same or opposite direction they are drawn parallel. In the diagram below, u v. u v Ex. If u 2 in the diagram below, draw 2 u u

3 Thus, multiplication of a vector by a scalar number results in a new vector parallel to the original but with a different magnitude. In general, two vectors u and v are parallel iff u = k v. AB BC AC If we go from point A to point B and then from point B to point C, the result is the same as going from point A to point C. B C A The angle between two vectors u and v is measured when the vectors are drawn tail to tail. The angle between them is always less than or equal to 180. u v Any vector v has a unit vector vˆ that has the same direction as v but a magnitude of 1. Recall that the magnitude of a vector is denoted by v. Therefore 1 v vˆ and v vˆ = v v A vector which has a magnitude of zero and thus a direction which is undefined (a point) is called the zero vector and is denoted by 0.

4 Ex. a) Given the rectangle ABCD, state: A E B 3 G H D F 8 C (i) 2 equal vectors (ii) 2 parallel vectors with different magnitudes (iii) AB as a scalar multiple of BE (iv) the angle between GC and EB b) Express AB in terms of: (i) AE (ii) FD Pg. 279 # 1 6, 8, 9, 10

5 MCV 4U Lesson 1.2 Vector Addition One of the primary tasks when working with vectors is finding the effect of two vectors or the sum of two vectors. The sum of two vectors has the same effect as the individual vectors. If the displacement vectors a and b move us two steps east and three steps south respectively, then the result of these two vectors will move us steps in the south of east where tan = 3 2. a b The sum of two vectors is called the resultant. From the diagram we can see that if we join the two vectors together by placing them tail to tip, then the resultant is the vector that starts at the first tail and extend to the last tip. The lengths and angles involved can be calculated using trigonometry. Triangle Law of Vector Addition Place the two vectors tip to tail and the resultant extends from the first tail to the last tip. a b a + b Parallelogram Law of Addition If two vectors are tail to tail we can find the resultant by using the vectors as two adjacent sides of a parallelogram. Draw the remaining two sides of the parallelogram and the resultant extends from the common tail point to the opposite corner of the parallelogram. a a + b b

6 Ex. 1 For the vectors below, find: a) u + v b) u v v u Ex. 2 Consider parallelogram EFGH with diagonals EG and FH that intersect at J. a) Express each vector as the sum of two other vectors in two ways. (i) HF (ii) FH (iii) GJ b) Express each vector as the difference of two other vectors in two ways. (i) HF (ii) FH (iii) GJ Ex. 3 In an orienteering race, you walk 100 m due east and the walk N70 E for 60 m. How far are you from your original position and in what direction?

7 Ex. 4 Find the resultant of the following parallel vectors. a) 5 km/h E 7 km/h E b) 30 km/h E 65 km/h W c) 50 N SW 20 N NE Pg. 290 # 3, 4, 6, 7, 11 14

8 MCV 4U Lesson 1.3 Multiplication of a Vector by a Scalar When you multiply a vector by a scalar, the magnitude of the vector is multiplied by the scalar and the vectors are parallel. If the scalar is positive, the direction remains unchanged. If the scalar is negative, the direction becomes opposite. ie: For the vector k a, where k is a scalar and a is a nonzero vector with magnitude a : If k 0, then k a has the same direction as a and has a magnitude of k a. If k 0, then k a has the opposite direction as a and has a magnitude of k a. Two vectors a and b are collinear iff it is possible to find a non-zero scalar k such that a kb. 1 a a is a unit vector,( vector with a length of 1) in the same direction as a. 1 a a is a unit vector,( vector with a length of 1) in the opposite direction as a. Linear combinations of vectors can be formed by adding scalar multiples of two or more vectors. ie: u 2a 3b Ex. 1 a) Which of the vectors below are scalar multiples of vector v? Explain. b) Find the value of the scalar k for each vector in (a). b d v f a

9 Ex. 2 Consider vector u with magnitude u = 100 km/h in a direction of N40 E. Draw a vector to represent each scalar multiple and describe the resulting vector. a) 3 u b) 0.5 u c) 2 u u Ex. 3 In trapezoid ABCD, BC AD and AD = 3BC. Let AB u and BC v. Express AD, BD, and CD as linear combinations of u and v. v u B C A D

10 Ex. 4 a and b are unit vectors that have an angle of 20 between them. a) Find the value of 3a 2b. b) Find the direction of 3a 2b. c) Find the unit vector in the same direction as 3a 2b. Pg. 298 # 4, 5, 7a, 11, 13, 15, 17, 19

11 MCV 4U Lesson 1.4 Vector Properties Properties of Vector Addition Commutative Law: Associative Law: a + b = b + a a + ( b + c ) = ( a + b ) + c Properties of Scalar Multiplication Associative Law: (mn) a = m(n a ) Distributive Law: m( a + b ) = m a + m b (m + n) a = m a + n a Properties of the Zero Vector a + 0 = a Each vector a has a negative, a, such that a + ( a ) = 0 Triangle Inequality For vectors a and b, a + b a b Since vectors a and b and their sum a + b form the sides of a triangle, the lengths of the sides of the triangle are the magnitudes of the vectors. From the figure 1 below, the side a + b must be less than the sum of the other two sides, a b, otherwise there is no triangle. Therefore a + b a b. When a and b have the same direction, then we have a case where a + b a b, as seen in figure 2. a + b a figure 1 b a a + b figure 2 b Ex. 1 Simplify. 2(u 2v 4w ) 3(2u v 3w )

12 Ex. 2 If a 2i 3 j k and b i 3k and c 3i 2 j k, find each of the following in terms of i, j, and k. a) a b b) 2a 3b c Pg. 306 # 3, 5, 7, 8ac, 9, 11

13 MCV 4U Lesson 1.5 Vectors in Two Space (R 2 ) and Three Space(R 3 ) By placing vectors on the Cartesian Plane, we can use algebraic methods in our study of vectors. 2 Dimensions (R 2 ) The ordered pair (a, b) is refered to as an algebraic vector. ie: OP = (a, b) The values of a and b are called the x- and y-components of the vector. Its tail is located at (0, 0) and its head is located at (a, b). N.B. (a, b) can represent a point with coordinates a and b, or it can represent a vector with components a and b. You should be able to tell by the context of a problem which is being meant. If you see an equal sign beside the brackets this would indicate a vector. ie: A(3, 4) is a point, while a = (3, 4) is a vector. 3 Dimensions (R 3 ) (a.k.a. 3 space or space) Similarly, any vector u in space can be written as an ordered triple where: u = OP = (a, b, c) where a, b, and c are its x-, y-, and z-components Its tail is located at (0, 0) and its head is located at (a, b, c). To plot a vector in space, you move a units along the x-axis, b units parallel to the y-axis and then c units parallel to the z-axis. Drawing a rectangular box (prism) is sometimes helpful with this In R 3, the three mutually perpendicular axes form a right handed system. Right Handed System In R 2 or R 3 the location of every point is unique. As a result,every vector drawn with its tail at the origin and its head at a point is also unique. These types of vec tors are known as position vectors.

14 Ex. 2 Locate the point P, and sketch the position of vector OP in three dimensions. a) (2, 0, 0) b) (0, 0, 3) c) (1, 2, 0) d) (2, 3, 4) e) (-3, 4, -2) f) (4, -5, 1) g) (0, -3, -6) h) (-4, -5, -2) z y x Ex. 2 What vector is represented in each of the following diagrams? Pg. 316 # 3, 5, 6, 7, 9, 11, 13, 14, 15

15

16 MCV 4U Lesson 1.6 Operations with Algebraic Vectors in Two Space (R 2 ) Unit Vectors in Two Dimensions Define vectors î = (1, 0) and ĵ = (0, 1). The vectors of length 1 that point in the direction of the positive x-axis and positive y-axis respectively. Vector u = (a, b) can also be represented as u = ai bj. For example, the vector OA (2, 5) can be written as OA 2i 5 j We can also use the unit vectors i and j to define a vector in the plane where i = (1, 0) a unit vector along the x-axis and j = (0, 1) a unit vector along the y-axis. y 1 (i, j ) j i 1 x any vector u can also be written in the form u = a i + b j Any vector u in a plane can be written as an ordered pair (a, b), where its magnitude u and direction are given by the equations : u = a 2 b 2 b and tan = a with being measured counter clockwise from the positive x axis to the vector. Vector Addition and Subtraction in Component Form: If u = (u 1, u 2 ) and v = (v 1, v 2 ), then u + v = (u 1 + v 1, u 2 + v 2 ) and similarly u v = (u 1 v 1, u 2 v 2 ). Scalar Multiplication in Component Form: If u = (u 1, u 2, u 3 ) and k R, then k u = (k u 1, k u 2 ) Remember: Two vectors are parallel or collinear iff one vector is a scalar multiple of the other. ie: If u kv for some scalar k, then u 1 v 1 u 2 v 2 k Components of a Vector Between Two Points: Given: P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) then P 1 P 2 (x 2 x 1, y 2 y 1 ) Proof: P 1 P 1 P 2 P 1 O OP 2 P 2 OP 2 OP 1 x, y ) ( x, ( y1 (x 2 x 1, y 2 y 1 ) ) O

17 Ex. 1 Given parallelogram ABCD with vertices A(1, 4), B(2, 3), C( 1, 4), find the coordinates of D. Ex. 2 Using vectors show that the points P(3, 7), Q(4, 8), and R(6, 10) are collinear.

18 Ex. 3 Given a 2i 3 j and b i 3 j determine: a) 2a 3b b) 2(3a b ) 2(a 2b ) c) 2a 3b Pg. 324 # 1, 3, 4, (5, 6)b, 8bd, 10, (13, 15)a

19 MCV 4U Lesson 1.7 Operations with Algebraic Vectors in Three Space (R 3 ) All vectors in space can be expressed in terms of the unit vectors i (1,0,0) and j (0,1,0) and k (0,0,1). These vectors are called basis vectors. The representation of a vector in terms of the basis vectors is unique. ie: If v a i + b j + c k, then there is only one set of values for a, b, and c. Thus, the expression of a vector in terms of a set of basis vectors is unique. If u = (a, b, c), then a, b, and c are the Cartesian components of u Its magnitude is given by u a b c Two vectors are equal iff their respective Cartesian coordinates are equal. Vector Addition and Subtraction in Component Form: If u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ), then u + v = (u 1 + v 1, u 2 + v 2, u 3 + v 3 ) and similarly u v = (u 1 v 1, u 2 v 2, u 3 v 3 ). Scalar Multiplication in Component Form: If u = (u 1, u 2, u 3 ) and k R, then k u = (k u 1, k u 2, k u 3 ) Remember: Two vectors are parallel or collinear iff one vector is a scalar multiple of the other. ie: If u kv for some scalar k, then u 1 v 1 u 2 v 2 u 3 v 3 k Components of a Vector Between Two Points: Given: P 1 (x 1, y 1, z 1 ) and P 2 (x 2, y 2, z 2 ) then P 1 P 2 (x 2 x 1, y 2 y 1, z 2 z 1 ) Proof: P 1 P 1 P 2 P 1 O OP 2 P 2 OP 2 OP 1 (x 2, y 2, z 2 ) (x 1, y 1, z 1 ) (x 2 x 1, y 2 y 1, z 2 z 1 ) The vector AB between two points with its tail at A( x 1, y 1,z 1 )A and head at B(x 2,y 2,z 2 ) is detemined AB OB OA (x 2 x 1, y 2 y 1, z 2 z 1 ).

20 Ex. 1 Given parallelogram ABCD with vertices A(1, 4, 6), B(2, 3, 7), C( 1, 4, 5), find the coordinates of D. Ex. 2 Given x i 3 j 2k and y 2i 3k, determine the following. a) 2x 4y b) 2x 4y Pg. 332 # 1, 2, 4, (5, 6)ac, 8, 9, 11, 12

21 MCV 4U Lesson 1.8 Linear Combinations & Spanning Sets Recall: Given a = (-1, 2) and b = (1, 4), find 2a 3b. The vector we created is a vector on the xy-plane and is the diagonal of the parallelogram formed by 2a and 3b. Linear Combinations of Vectors: Given noncollinear vectors u and v, a linear combination of these vectors is au + bv, where a and b are scalars. Spanning Sets: Any collinear Vectors span R 1. Any non-collinear, nonzero vectors span R 2. i.e. if you can write the vectors as a linear combination of each other, they create a spanning set in R 2. Since every vector is R 2 can be written as a linear combination of the vectors i a spanning set in R 2. and j, they form

22 Ex. 1 Show that = (4, 23) can be written as a linear combination of the vectors ( 1,4) and (2,5). Ex. 2 Do (3,4) and (9,12) span R 2?

23 This concept can be extended in R 3., and form a basis for R 3. Therefore, every vector in R 3 can be written as a linear combination of these three vectors. Spanning sets in R 3 : Any pair of nonzero, noncollinear vectors span a plane in 3-space. i.e. if you can write them as a linear combination, then they span a plane in R 3. Ex. 3 Does vector ( 9, 4,1) lie in the same plane as ( 1, 2,1) and (3, 1,1)? ie: Do they span a plane in R 3? In general, when we are trying to determine whether a vector lies in the plane determined by two other nonzero, noncollinear vectors, it is sufficient to solve any pair of equations and look for consistency in the third equation. If the result is consistent, the vector lies in the plane and the vectors span the plane, and if not, the vector does not lie in the plane. Ex. 4 Show that the vectors ( 1, 2, 3), ( 4,1, 2) and ( 14, 1,16) do not lie on the same plane. Pg. 340 #1,3, 5, 7b, 9bd, 11, 12, 13b, 14, 15

Review of Coordinate Systems

Review of Coordinate Systems Vector in 2 R and 3 R 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

More information

Coordinate Systems. Chapter 3. Cartesian Coordinate System. Polar Coordinate System

Coordinate Systems. Chapter 3. Cartesian Coordinate System. Polar Coordinate System Chapter 3 Vectors Coordinate Systems Used to describe the position of a point in space Coordinate system consists of a fixed reference point called the origin specific axes with scales and labels instructions

More information

Main Ideas in Class Today

Main Ideas in Class Today Main Ideas in Class Today After today, you should be able to: Understand vector notation Use basic trigonometry in order to find the x and y components of a vector (only right triangles) Add and subtract

More information

Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems

Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems To locate a point in a plane, two numbers are necessary. We know that any point in the plane can be represented as an ordered pair (a, b) of real numbers, where a is the x-coordinate and b is the y-coordinate.

More information

Vectors. Introduction. Prof Dr Ahmet ATAÇ

Vectors. Introduction. Prof Dr Ahmet ATAÇ Chapter 3 Vectors Vectors Vector quantities Physical quantities that have both n u m e r i c a l a n d d i r e c t i o n a l properties Mathematical operations of vectors in this chapter A d d i t i o

More information

Quantities which have only magnitude are called scalars. Quantities which have magnitude and direction are called vectors.

Quantities which have only magnitude are called scalars. Quantities which have magnitude and direction are called vectors. Vectors summary Quantities which have only magnitude are called scalars. Quantities which have magnitude and direction are called vectors. AB is the position vector of B relative to A and is the vector

More information

DATE: MATH ANALYSIS 2 CHAPTER 12: VECTORS & DETERMINANTS

DATE: MATH ANALYSIS 2 CHAPTER 12: VECTORS & DETERMINANTS NAME: PERIOD: DATE: MATH ANALYSIS 2 MR. MELLINA CHAPTER 12: VECTORS & DETERMINANTS Sections: v 12.1 Geometric Representation of Vectors v 12.2 Algebraic Representation of Vectors v 12.3 Vector and Parametric

More information

VECTORS. Vectors OPTIONAL - I Vectors and three dimensional Geometry

VECTORS. Vectors OPTIONAL - I Vectors and three dimensional Geometry Vectors OPTIONAL - I 32 VECTORS In day to day life situations, we deal with physical quantities such as distance, speed, temperature, volume etc. These quantities are sufficient to describe change of position,

More information

Definition: A vector is a directed line segment which represents a displacement from one point P to another point Q.

Definition: A vector is a directed line segment which represents a displacement from one point P to another point Q. THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH Algebra Section : - Introduction to Vectors. You may have already met the notion of a vector in physics. There you will have

More information

Part (1) Second : Trigonometry. Tan

Part (1) Second : Trigonometry. Tan Part (1) Second : Trigonometry (1) Complete the following table : The angle Ratio 42 12 \ Sin 0.3214 Cas 0.5321 Tan 2.0625 (2) Complete the following : 1) 46 36 \ 24 \\ =. In degrees. 2) 44.125 = in degrees,

More information

Prepared by: M. S. KumarSwamy, TGT(Maths) Page

Prepared by: M. S. KumarSwamy, TGT(Maths) Page Prepared by: M S KumarSwamy, TGT(Maths) Page - 119 - CHAPTER 10: VECTOR ALGEBRA QUICK REVISION (Important Concepts & Formulae) MARKS WEIGHTAGE 06 marks Vector The line l to the line segment AB, then a

More information

Physics 40 Chapter 3: Vectors

Physics 40 Chapter 3: Vectors Physics 40 Chapter 3: Vectors 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

More information

(arrows denote positive direction)

(arrows denote positive direction) 12 Chapter 12 12.1 3-dimensional Coordinate System The 3-dimensional coordinate system we use are coordinates on R 3. The coordinate is presented as a triple of numbers: (a,b,c). In the Cartesian coordinate

More information

Vectors. Introduction

Vectors. Introduction Chapter 3 Vectors Vectors Vector quantities Physical quantities that have both numerical and directional properties Mathematical operations of vectors in this chapter Addition Subtraction Introduction

More information

Day 1: Introduction to Vectors + Vector Arithmetic

Day 1: Introduction to Vectors + Vector Arithmetic Day 1: Introduction to Vectors + Vector Arithmetic A is a quantity that has magnitude but no direction. You can have signed scalar quantities as well. A is a quantity that has both magnitude and direction.

More information

MAT 1339-S14 Class 8

MAT 1339-S14 Class 8 MAT 1339-S14 Class 8 July 28, 2014 Contents 7.2 Review Dot Product........................... 2 7.3 Applications of the Dot Product..................... 4 7.4 Vectors in Three-Space.........................

More information

General Physics I, Spring Vectors

General Physics I, Spring Vectors General Physics I, Spring 2011 Vectors 1 Vectors: Introduction A vector quantity in physics is one that has a magnitude (absolute value) and a direction. We have seen three already: displacement, velocity,

More information

Chapter 3 Vectors Prof. Raymond Lee, revised

Chapter 3 Vectors Prof. Raymond Lee, revised Chapter 3 Vectors Prof. Raymond Lee, revised 9-2-2010 1 Coordinate systems Used to describe a point s position in space Coordinate system consists of fixed reference point called origin specific axes with

More information

Physics 20 Lesson 10 Vector Addition

Physics 20 Lesson 10 Vector Addition Physics 20 Lesson 10 Vector Addition I. Vector Addition in One Dimension (It is strongly recommended that you read pages 70 to 75 in Pearson for a good discussion on vector addition in one dimension.)

More information

2- Scalars and Vectors

2- Scalars and Vectors 2- Scalars and Vectors Scalars : have magnitude only : Length, time, mass, speed and volume is example of scalar. v Vectors : have magnitude and direction. v The magnitude of is written v v Position, displacement,

More information

Vectors 1. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.

Vectors 1. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Vectors 1 The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Launch Mathematica. Type

More information

Section 8.4 Vector and Parametric Equations of a Plane

Section 8.4 Vector and Parametric Equations of a Plane Section 8.4 Vector and Parametric Equations of a Plane In the previous section, the vector, parametric, and symmetric equations of lines in R 3 were developed. In this section, we will develop vector and

More information

Vectors and the Geometry of Space

Vectors and the Geometry of Space Vectors and the Geometry of Space Many quantities in geometry and physics, such as area, volume, temperature, mass, and time, can be characterized by a single real number scaled to appropriate units of

More information

9.1. Basic Concepts of Vectors. Introduction. Prerequisites. Learning Outcomes. Learning Style

9.1. Basic Concepts of Vectors. Introduction. Prerequisites. Learning Outcomes. Learning Style Basic Concepts of Vectors 9.1 Introduction In engineering, frequent reference is made to physical quantities, such as force, speed and time. For example, we talk of the speed of a car, and the force in

More information

CHAPTER 10 VECTORS POINTS TO REMEMBER

CHAPTER 10 VECTORS POINTS TO REMEMBER For more important questions visit : www4onocom CHAPTER 10 VECTORS POINTS TO REMEMBER A quantity that has magnitude as well as direction is called a vector It is denoted by a directed line segment Two

More information

Mathematical review trigonometry vectors Motion in one dimension

Mathematical review trigonometry vectors Motion in one dimension Mathematical review trigonometry vectors Motion in one dimension Used to describe the position of a point in space Coordinate system (frame) consists of a fixed reference point called the origin specific

More information

Chapter 2 - Vector Algebra

Chapter 2 - Vector Algebra A spatial vector, or simply vector, is a concept characterized by a magnitude and a direction, and which sums with other vectors according to the Parallelogram Law. A vector can be thought of as an arrow

More information

Chapter 8 Vectors and Scalars

Chapter 8 Vectors and Scalars Chapter 8 193 Vectors and Scalars Chapter 8 Vectors and Scalars 8.1 Introduction: In this chapter we shall use the ideas of the plane to develop a new mathematical concept, vector. If you have studied

More information

Vectors and 2D Kinematics. AIT AP Physics C

Vectors and 2D Kinematics. AIT AP Physics C Vectors and 2D Kinematics Coordinate Systems Used to describe the position of a point in space Coordinate system consists of a fixed reference point called the origin specific axes with scales and labels

More information

10.1 Vectors. c Kun Wang. Math 150, Fall 2017

10.1 Vectors. c Kun Wang. Math 150, Fall 2017 10.1 Vectors Definition. A vector is a quantity that has both magnitude and direction. A vector is often represented graphically as an arrow where the direction is the direction of the arrow, and the magnitude

More information

P1 Chapter 11 :: Vectors

P1 Chapter 11 :: Vectors P1 Chapter 11 :: Vectors jfrost@tiffin.kingston.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 21 st August 2017 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework

More information

Introduction to Vectors

Introduction to Vectors Introduction to Vectors Why Vectors? Say you wanted to tell your friend that you re running late and will be there in five minutes. That s precisely enough information for your friend to know when you

More information

Detailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors

Detailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors Unit 1 Vectors In this unit, we introduce vectors, vector operations, and equations of lines and planes. Note: Unit 1 is based on Chapter 12 of the textbook, Salas and Hille s Calculus: Several Variables,

More information

Unit 1 Representing and Operations with Vectors. Over the years you have come to accept various mathematical concepts or properties:

Unit 1 Representing and Operations with Vectors. Over the years you have come to accept various mathematical concepts or properties: Lesson1.notebook November 27, 2012 Algebra Unit 1 Representing and Operations with Vectors Over the years you have come to accept various mathematical concepts or properties: Communative Property Associative

More information

A FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION

A FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler R n : Vectors Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC BY-NC-SA) This

More information

9.4 Polar Coordinates

9.4 Polar Coordinates 9.4 Polar Coordinates Polar coordinates uses distance and direction to specify a location in a plane. The origin in a polar system is a fixed point from which a ray, O, is drawn and we call the ray the

More information

Unit 8. ANALYTIC GEOMETRY.

Unit 8. ANALYTIC GEOMETRY. Unit 8. ANALYTIC GEOMETRY. 1. VECTORS IN THE PLANE A vector is a line segment running from point A (tail) to point B (head). 1.1 DIRECTION OF A VECTOR The direction of a vector is the direction of the

More information

SECTION 6.3: VECTORS IN THE PLANE

SECTION 6.3: VECTORS IN THE PLANE (Section 6.3: Vectors in the Plane) 6.18 SECTION 6.3: VECTORS IN THE PLANE Assume a, b, c, and d are real numbers. PART A: INTRO A scalar has magnitude but not direction. We think of real numbers as scalars,

More information

11.1 Vectors in the plane

11.1 Vectors in the plane 11.1 Vectors in the plane What is a vector? It is an object having direction and length. Geometric way to represent vectors It is represented by an arrow. The direction of the arrow is the direction of

More information

Congruence Axioms. Data Required for Solving Oblique Triangles

Congruence Axioms. Data Required for Solving Oblique Triangles Math 335 Trigonometry Sec 7.1: Oblique Triangles and the Law of Sines In section 2.4, we solved right triangles. We now extend the concept to all triangles. Congruence Axioms Side-Angle-Side SAS Angle-Side-Angle

More information

2.1 Scalars and Vectors

2.1 Scalars and Vectors 2.1 Scalars and Vectors Scalar A quantity characterized by a positive or negative number Indicated by letters in italic such as A e.g. Mass, volume and length 2.1 Scalars and Vectors Vector A quantity

More information

VECTORS vectors & scalars vector direction magnitude scalar only magnitude

VECTORS vectors & scalars vector direction magnitude scalar only magnitude VECTORS Physical quantities are classified in two big classes: vectors & scalars. A vector is a physical quantity which is completely defined once we know precisely its direction and magnitude (for example:

More information

Vectors. Examples of vectors include: displacement, velocity, acceleration, and force. Examples of scalars include: distance, speed, time, and volume.

Vectors. Examples of vectors include: displacement, velocity, acceleration, and force. Examples of scalars include: distance, speed, time, and volume. Math 150 Prof. Beydler 7.4/7.5 Notes Page 1 of 6 Vectors Suppose a car is heading NE (northeast) at 60 mph. We can use a vector to help draw a picture (see right). v A vector consists of two parts: 1.

More information

The Dot Product Pg. 377 # 6ace, 7bdf, 9, 11, 14 Pg. 385 # 2, 3, 4, 6bd, 7, 9b, 10, 14 Sept. 25

The Dot Product Pg. 377 # 6ace, 7bdf, 9, 11, 14 Pg. 385 # 2, 3, 4, 6bd, 7, 9b, 10, 14 Sept. 25 UNIT 2 - APPLICATIONS OF VECTORS Date Lesson TOPIC Homework Sept. 19 2.1 (11) 7.1 Vectors as Forces Pg. 362 # 2, 5a, 6, 8, 10 13, 16, 17 Sept. 21 2.2 (12) 7.2 Velocity as Vectors Pg. 369 # 2,3, 4, 6, 7,

More information

8.0 Definition and the concept of a vector:

8.0 Definition and the concept of a vector: Chapter 8: Vectors In this chapter, we will study: 80 Definition and the concept of a ector 81 Representation of ectors in two dimensions (2D) 82 Representation of ectors in three dimensions (3D) 83 Operations

More information

Vector Algebra August 2013

Vector Algebra August 2013 Vector Algebra 12.1 12.2 28 August 2013 What is a Vector? A vector (denoted or v) is a mathematical object possessing both: direction and magnitude also called length (denoted ). Vectors are often represented

More information

9.2. Cartesian Components of Vectors. Introduction. Prerequisites. Learning Outcomes

9.2. Cartesian Components of Vectors. Introduction. Prerequisites. Learning Outcomes Cartesian Components of Vectors 9.2 Introduction It is useful to be able to describe vectors with reference to specific coordinate sstems, such as the Cartesian coordinate sstem. So, in this Section, we

More information

Vectors. The standard geometric definition of vector is as something which has direction and magnitude but not position.

Vectors. The standard geometric definition of vector is as something which has direction and magnitude but not position. Vectors The standard geometric definition of vector is as something which has direction and magnitude but not position. Since vectors have no position we may place them wherever is convenient. Vectors

More information

chapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?

chapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true? chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " "

More information

Chapter 2: Force Vectors

Chapter 2: Force Vectors Chapter 2: Force Vectors Chapter Objectives To show how to add forces and resolve them into components using the Parallelogram Law. To express force and position in Cartesian vector form and explain how

More information

Chapter 3. Vectors. 3.1 Coordinate Systems 3.2 Vector and Scalar Quantities 3.3 Some Properties of Vectors 3.4 Components of a Vector and Unit Vectors

Chapter 3. Vectors. 3.1 Coordinate Systems 3.2 Vector and Scalar Quantities 3.3 Some Properties of Vectors 3.4 Components of a Vector and Unit Vectors Chapter 3 Vectors 3.1 Coordinate Systems 3.2 Vector and Scalar Quantities 3.3 Some Properties of Vectors 3.4 Components of a Vector and Unit Vectors 1 Vectors Vector quantities Physical quantities that

More information

Course Notes Math 275 Boise State University. Shari Ultman

Course Notes Math 275 Boise State University. Shari Ultman Course Notes Math 275 Boise State University Shari Ultman Fall 2017 Contents 1 Vectors 1 1.1 Introduction to 3-Space & Vectors.............. 3 1.2 Working With Vectors.................... 7 1.3 Introduction

More information

FORCE TABLE INTRODUCTION

FORCE TABLE INTRODUCTION FORCE TABLE INTRODUCTION All measurable quantities can be classified as either a scalar 1 or a vector 2. A scalar has only magnitude while a vector has both magnitude and direction. Examples of scalar

More information

Worksheet 1.1: Introduction to Vectors

Worksheet 1.1: Introduction to Vectors Boise State Math 275 (Ultman) Worksheet 1.1: Introduction to Vectors From the Toolbox (what you need from previous classes) Know how the Cartesian coordinates a point in the plane (R 2 ) determine its

More information

CONTENTS. INTRODUCTION MEQ curriculum objectives for vectors (8% of year). page 2 What is a vector? What is a scalar? page 3, 4

CONTENTS. INTRODUCTION MEQ curriculum objectives for vectors (8% of year). page 2 What is a vector? What is a scalar? page 3, 4 CONTENTS INTRODUCTION MEQ crriclm objectives for vectors (8% of year). page 2 What is a vector? What is a scalar? page 3, 4 VECTOR CONCEPTS FROM GEOMETRIC AND ALGEBRAIC PERSPECTIVES page 1 Representation

More information

Unit 1: Math Toolbox Math Review Guiding Light #1

Unit 1: Math Toolbox Math Review Guiding Light #1 Unit 1: Math Toolbox Math Review Guiding Light #1 Academic Physics Unit 1: Math Toolbox Math Review Guiding Light #1 Table of Contents Topic Slides Algebra Review 2 8 Trigonometry Review 9 16 Scalar &

More information

3 Vectors. 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan

3 Vectors. 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan Chapter 3 Vectors 3 Vectors 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 2 3 3-2 Vectors and Scalars Physics deals with many quantities that have both size and direction. It needs a special mathematical

More information

1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4

1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4 MATH2202 Notebook 1 Fall 2015/2016 prepared by Professor Jenny Baglivo Contents 1 MATH2202 Notebook 1 3 1.1 Single Variable Calculus versus Multivariable Calculus................... 3 1.2 Rectangular Coordinate

More information

LINEAR ALGEBRA - CHAPTER 1: VECTORS

LINEAR ALGEBRA - CHAPTER 1: VECTORS LINEAR ALGEBRA - CHAPTER 1: VECTORS A game to introduce Linear Algebra In measurement, there are many quantities whose description entirely rely on magnitude, i.e., length, area, volume, mass and temperature.

More information

Physics 12. Chapter 1: Vector Analysis in Two Dimensions

Physics 12. Chapter 1: Vector Analysis in Two Dimensions Physics 12 Chapter 1: Vector Analysis in Two Dimensions 1. Definitions When studying mechanics in Physics 11, we have realized that there are two major types of quantities that we can measure for the systems

More information

CE 201 Statics. 2 Physical Sciences. Rigid-Body Deformable-Body Fluid Mechanics Mechanics Mechanics

CE 201 Statics. 2 Physical Sciences. Rigid-Body Deformable-Body Fluid Mechanics Mechanics Mechanics CE 201 Statics 2 Physical Sciences Branch of physical sciences 16 concerned with the state of Mechanics rest motion of bodies that are subjected to the action of forces Rigid-Body Deformable-Body Fluid

More information

Phys 221. Chapter 3. Vectors A. Dzyubenko Brooks/Cole

Phys 221. Chapter 3. Vectors A. Dzyubenko Brooks/Cole Phs 221 Chapter 3 Vectors adzubenko@csub.edu http://www.csub.edu/~adzubenko 2014. Dzubenko 2014 rooks/cole 1 Coordinate Sstems Used to describe the position of a point in space Coordinate sstem consists

More information

For more information visit here:

For more information visit here: The length or the magnitude of the vector = (a, b, c) is defined by w = a 2 +b 2 +c 2 A vector may be divided by its own length to convert it into a unit vector, i.e.? = u / u. (The vectors have been denoted

More information

Ground Rules. PC1221 Fundamentals of Physics I. Coordinate Systems. Cartesian Coordinate System. Lectures 5 and 6 Vectors.

Ground Rules. PC1221 Fundamentals of Physics I. Coordinate Systems. Cartesian Coordinate System. Lectures 5 and 6 Vectors. PC1221 Fundamentals of Phsics I Lectures 5 and 6 Vectors Dr Ta Seng Chuan 1 Ground ules Switch off our handphone and pager Switch off our laptop computer and keep it No talking while lecture is going on

More information

CfE Higher Mathematics Course Materials Topic 2: Vectors

CfE Higher Mathematics Course Materials Topic 2: Vectors SCHOLAR Study Guide CfE Higher Mathematics Course Materials Topic : Vectors Authored by: Margaret Ferguson Reviewed by: Jillian Hornby Previously authored by: Jane S Paterson Dorothy A Watson Heriot-Watt

More information

UNIT 1 VECTORS INTRODUCTION 1.1 OBJECTIVES. Stucture

UNIT 1 VECTORS INTRODUCTION 1.1 OBJECTIVES. Stucture UNIT 1 VECTORS 1 Stucture 1.0 Introduction 1.1 Objectives 1.2 Vectors and Scalars 1.3 Components of a Vector 1.4 Section Formula 1.5 nswers to Check Your Progress 1.6 Summary 1.0 INTRODUCTION In this unit,

More information

Test Corrections for Unit 1 Test

Test Corrections for Unit 1 Test MUST READ DIRECTIONS: Read the directions located on www.koltymath.weebly.com to understand how to properly do test corrections. Ask for clarification from your teacher if there are parts that you are

More information

Welcome to IB Math - Standard Level Year 2

Welcome to IB Math - Standard Level Year 2 Welcome to IB Math - Standard Level Year 2 Why math? Not So Some things to know: Good HW Good HW Good HW www.aleimath.blogspot.com Example 1. Lots of info at Example Example 2. HW yup. You know you love

More information

AP Physics C Mechanics Vectors

AP Physics C Mechanics Vectors 1 AP Physics C Mechanics Vectors 2015 12 03 www.njctl.org 2 Scalar Versus Vector A scalar has only a physical quantity such as mass, speed, and time. A vector has both a magnitude and a direction associated

More information

b) The trend is for the average slope at x = 1 to decrease. The slope at x = 1 is 1.

b) The trend is for the average slope at x = 1 to decrease. The slope at x = 1 is 1. Chapters 1 to 8 Course Review Chapters 1 to 8 Course Review Question 1 Page 509 a) i) ii) [2(16) 12 + 4][2 3+ 4] 4 1 [2(2.25) 4.5+ 4][2 3+ 4] 1.51 = 21 3 = 7 = 1 0.5 = 2 [2(1.21) 3.3+ 4][2 3+ 4] iii) =

More information

Mathematics Revision Guides Vectors Page 1 of 19 Author: Mark Kudlowski M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier VECTORS

Mathematics Revision Guides Vectors Page 1 of 19 Author: Mark Kudlowski M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier VECTORS Mathematics Revision Guides Vectors Page of 9 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier VECTORS Version:.4 Date: 05-0-05 Mathematics Revision Guides Vectors Page of 9 VECTORS

More information

Chapter 2 Mechanical Equilibrium

Chapter 2 Mechanical Equilibrium Chapter 2 Mechanical Equilibrium I. Force (2.1) A. force is a push or pull 1. A force is needed to change an object s state of motion 2. State of motion may be one of two things a. At rest b. Moving uniformly

More information

Notes: Review of Algebra I skills

Notes: Review of Algebra I skills Notes: Review of Algebra I skills http://www.monroeps.org/honors_geometry.aspx http://www.parklandsd.org/wp-content/uploads/hrs_geometry.pdf Name: Date: Period: Algebra Review: Systems of Equations * If

More information

ANALYTICAL GEOMETRY Revision of Grade 10 Analytical Geometry

ANALYTICAL GEOMETRY Revision of Grade 10 Analytical Geometry ANALYTICAL GEOMETRY Revision of Grade 10 Analtical Geometr Let s quickl have a look at the analtical geometr ou learnt in Grade 10. 8 LESSON Midpoint formula (_ + 1 ;_ + 1 The midpoint formula is used

More information

3.1 Using Vectors 3.3 Coordinate Systems and Vector Components.notebook September 19, 2017

3.1 Using Vectors 3.3 Coordinate Systems and Vector Components.notebook September 19, 2017 Using Vectors A vector is a quantity with both a size (magnitude) and a direction. Figure 3.1 shows how to represent a particle s velocity as a vector. Section 3.1 Using Vectors The particle s speed at

More information

CHAPTER 4 VECTORS. Before we go any further, we must talk about vectors. They are such a useful tool for

CHAPTER 4 VECTORS. Before we go any further, we must talk about vectors. They are such a useful tool for CHAPTER 4 VECTORS Before we go any further, we must talk about vectors. They are such a useful tool for the things to come. The concept of a vector is deeply rooted in the understanding of physical mechanics

More information

Kinematics in Two Dimensions; Vectors

Kinematics in Two Dimensions; Vectors Kinematics in Two Dimensions; Vectors Vectors & Scalars!! Scalars They are specified only by a number and units and have no direction associated with them, such as time, mass, and temperature.!! Vectors

More information

Fundamental Electromagnetics ( Chapter 2: Vector Algebra )

Fundamental Electromagnetics ( Chapter 2: Vector Algebra ) Fundamental Electromagnetics ( Chapter 2: Vector Algebra ) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 02-760-4253 Fax:02-6919-2160 1 Key Points Basic concept of scalars and vectors What is unit vector?

More information

Distance Formula in 3-D Given any two points P 1 (x 1, y 1, z 1 ) and P 2 (x 2, y 2, z 2 ) the distance between them is ( ) ( ) ( )

Distance Formula in 3-D Given any two points P 1 (x 1, y 1, z 1 ) and P 2 (x 2, y 2, z 2 ) the distance between them is ( ) ( ) ( ) Vectors and the Geometry of Space Vector Space The 3-D coordinate system (rectangular coordinates ) is the intersection of three perpendicular (orthogonal) lines called coordinate axis: x, y, and z. Their

More information

When two letters name a vector, the first indicates the and the second indicates the of the vector.

When two letters name a vector, the first indicates the and the second indicates the of the vector. 8-8 Chapter 8 Applications of Trigonometry 8.3 Vectors, Operations, and the Dot Product Basic Terminology Algeraic Interpretation of Vectors Operations with Vectors Dot Product and the Angle etween Vectors

More information

2-9. The plate is subjected to the forces acting on members A and B as shown. If θ = 60 o, determine the magnitude of the resultant of these forces

2-9. The plate is subjected to the forces acting on members A and B as shown. If θ = 60 o, determine the magnitude of the resultant of these forces 2-9. The plate is subjected to the forces acting on members A and B as shown. If θ 60 o, determine the magnitude of the resultant of these forces and its direction measured clockwise from the positie x

More information

Chapter 13: Vectors and the Geometry of Space

Chapter 13: Vectors and the Geometry of Space Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic

More information

Chapter 13: Vectors and the Geometry of Space

Chapter 13: Vectors and the Geometry of Space Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic

More information

R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member

R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Chapter R Review of basic concepts * R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Ex: Write the set of counting numbers

More information

Vector components and motion

Vector components and motion Vector components and motion Objectives Distinguish between vectors and scalars and give examples of each. Use vector diagrams to interpret the relationships among vector quantities such as force and acceleration.

More information

Vectors. both a magnitude and a direction. Slide Pearson Education, Inc.

Vectors. both a magnitude and a direction. Slide Pearson Education, Inc. Vectors A quantity that is fully described The velocity vector has both a magnitude and a direction. by a single number is called a scalar quantity (i.e., mass, temperature, volume). A quantity having

More information

Investigation Find the area of the triangle. (See student text.)

Investigation Find the area of the triangle. (See student text.) Selected ACE: Looking For Pythagoras Investigation 1: #20, #32. Investigation 2: #18, #38, #42. Investigation 3: #8, #14, #18. Investigation 4: #12, #15, #23. ACE Problem Investigation 1 20. Find the area

More information

Chapter 2 Statics of Particles. Resultant of Two Forces 8/28/2014. The effects of forces on particles:

Chapter 2 Statics of Particles. Resultant of Two Forces 8/28/2014. The effects of forces on particles: Chapter 2 Statics of Particles The effects of forces on particles: - replacing multiple forces acting on a particle with a single equivalent or resultant force, - relations between forces acting on a particle

More information

Quiz No. 1: Tuesday Jan. 31. Assignment No. 2, due Thursday Feb 2: Problems 8.4, 8.13, 3.10, 3.28 Conceptual questions: 8.1, 3.6, 3.12, 3.

Quiz No. 1: Tuesday Jan. 31. Assignment No. 2, due Thursday Feb 2: Problems 8.4, 8.13, 3.10, 3.28 Conceptual questions: 8.1, 3.6, 3.12, 3. Quiz No. 1: Tuesday Jan. 31 Assignment No. 2, due Thursday Feb 2: Problems 8.4, 8.13, 3.10, 3.28 Conceptual questions: 8.1, 3.6, 3.12, 3.20 Chapter 3 Vectors and Two-Dimensional Kinematics Properties of

More information

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component

More information

Created by T. Madas 2D VECTORS. Created by T. Madas

Created by T. Madas 2D VECTORS. Created by T. Madas 2D VECTORS Question 1 (**) Relative to a fixed origin O, the point A has coordinates ( 2, 3). The point B is such so that AB = 3i 7j, where i and j are mutually perpendicular unit vectors lying on the

More information

Scalar & Vector tutorial

Scalar & Vector tutorial Scalar & Vector tutorial scalar vector only magnitude, no direction both magnitude and direction 1-dimensional measurement of quantity not 1-dimensional time, mass, volume, speed temperature and so on

More information

UNIT-05 VECTORS. 3. Utilize the characteristics of two or more vectors that are concurrent, or collinear, or coplanar.

UNIT-05 VECTORS. 3. Utilize the characteristics of two or more vectors that are concurrent, or collinear, or coplanar. UNIT-05 VECTORS Introduction: physical quantity that can be specified by just a number the magnitude is known as a scalar. In everyday life you deal mostly with scalars such as time, temperature, length

More information

Department of Physics, Korea University

Department of Physics, Korea University Name: Department: Notice +2 ( 1) points per correct (incorrect) answer. No penalty for an unanswered question. Fill the blank ( ) with (8) if the statement is correct (incorrect).!!!: corrections to an

More information

Pre-Calculus Vectors

Pre-Calculus Vectors Slide 1 / 159 Slide 2 / 159 Pre-Calculus Vectors 2015-03-24 www.njctl.org Slide 3 / 159 Table of Contents Intro to Vectors Converting Rectangular and Polar Forms Operations with Vectors Scalar Multiples

More information

[ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of

[ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of . Matrices A matrix is any rectangular array of numbers. For example 3 5 6 4 8 3 3 is 3 4 matrix, i.e. a rectangular array of numbers with three rows four columns. We usually use capital letters for matrices,

More information

Vectors. A vector is usually denoted in bold, like vector a, or sometimes it is denoted a, or many other deviations exist in various text books.

Vectors. A vector is usually denoted in bold, like vector a, or sometimes it is denoted a, or many other deviations exist in various text books. Vectors A Vector has Two properties Magnitude and Direction. That s a weirder concept than you think. A Vector does not necessarily start at a given point, but can float about, but still be the SAME vector.

More information

Fundamental Electromagnetics [ Chapter 2: Vector Algebra ]

Fundamental Electromagnetics [ Chapter 2: Vector Algebra ] Fundamental Electromagnetics [ Chapter 2: Vector Algebra ] Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 02-760-4253 Fax:02-6919-2160 1 Key Points Basic concept of scalars and vectors What is unit vector?

More information

St Andrew s Academy Mathematics Department Higher Mathematics

St Andrew s Academy Mathematics Department Higher Mathematics St Andrew s Academy Mathematics Department Higher Mathematics VECTORS hsn.uk.net Higher Mathematics Vectors Contents Vectors 1 1 Vectors and Scalars EF 1 Components EF 1 3 Magnitude EF 3 4 Equal Vectors

More information

Please Visit us at:

Please Visit us at: IMPORTANT QUESTIONS WITH ANSWERS Q # 1. Differentiate among scalars and vectors. Scalars Vectors (i) The physical quantities that are completely (i) The physical quantities that are completely described

More information