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DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 3. Geometry & Trigonometry / 3.9 Vector Properties / 3.9.1 Introduction to Vectors


3.9.1 Introduction to Vectors


Scalars & Vectors

What are scalars?

  • Scalars are quantities without direction
    • They have only a size (magnitude)
    • For example: speed, distance, time, mass
  • Most scalar quantities can never be negative
    • You cannot have a negative speed or distance

What are vectors?

  • Vectors are quantities which also have a direction, this is what makes them more than just a scalar
    • For example: two objects with velocities of 7 m/s and ‑7 m/s are travelling at the same speed but in opposite directions
  • A vector quantity is described by both its magnitude and direction
  • A vector has components in the direction of the x- , y-, and z- axes
    • Vector quantities can have positive or negative components
  • Some examples of vector quantities you may come across are displacement, velocity, acceleration, force/weight, momentum
    • Displacement is the position of an object from a starting point
    • Velocity is a speed in a given direction (displacement over time)
    • Acceleration is the change in velocity over time
  • Vectors may be given in either 2- or 3- dimensions

1.1.1 Scalars _ Vectors Diagram 1

Exam Tip

  • Make sure you fully understand the definitions of all the words in this section so that you can be clear about what your exam question is asking of you

Worked Example

State whether each of the following is a scalar or a vector quantity.

a)
A speed boat travels at 3 m/s on a bearing of 052°

3-9-1-ib-aa-hl-scalars--vectors-we-solution-a-

b)
A garden is 1.7 m wide

3-9-1-ib-aa-hl-scalars--vectors-we-solution-b-

c)
A car accelerates forwards at 5.4 ms-2

3-9-1-ib-aa-hl-scalars--vectors-we-solution-c-

d)
A film lasts 2 hours 17 minutes

3-9-1-ib-aa-hl-scalars--vectors-we-solution-d-

e)
An athlete runs at an average speed of 10.44 ms-1

3-9-1-ib-aa-hl-scalars--vectors-we-solution-e-

f)
A ball rolls forwards 60 cm before stopping

3-9-1-ib-aa-hl-scalars--vectors-we-solution-f-

 

Vector Notation

How are vectors represented?

  • Vectors are usually represented using an arrow in the direction of movement
    • The length of the arrow represents its magnitude
  • They are written as lowercase letters either in bold or underlined
    • For example a vector from the point O to A will be written a or a
      • The vector from the point A to O will be written -a or -a
  • If the start and end point of the vector is known, it is written using these points as capital letters with an arrow showing the direction of movement
    • For example: AB with rightwards arrow on top or BA with rightwards arrow on top
  • Two vectors are equal only if their corresponding components are equal
  • Numerically, vectors are either represented using column vectors or base vectors
    • Unless otherwise indicated, you may carry out all working and write your answers in either of these two types of vector notation

 

What are column vectors?

  • Column vectors are where one number is written above the other enclosed in brackets
  • In 2-dimensions the top number represents movement in the horizontal direction (right/left) and the bottom number represents movement in the vertical direction (up/down)
  • A positive value represents movement in the positive direction (right/up) and a negative value represents movement in the negative direction (left/down)
    • For example: The column vector begin mathsize 16px style open parentheses fraction numerator 3 over denominator negative 2 end fraction close parentheses end style represents 3 units in the positive horizontal (x) direction (i.e., right) and 2 units in the negative vertical (y) direction (i.e., down)
  • In 3-dimensions the top number represents the movement in the x direction (length), the middle number represents movement in the y direction (width) and the bottom number represents the movement in the z direction (depth)
    • For example: The column vector begin mathsize 16px style stretchy left parenthesis fraction numerator space space space 3
minus 4 over denominator space space space 2 end fraction stretchy right parenthesis end style
 represents 3 units in the positive x direction, 4 units in the negative y direction and 2 units in the positive z direction

What are base vectors?

  • Base vectors use i, j and k notation where i, j and k are unit vectors in the positive x, y, and z directions respectively
    • This is sometimes also called unit vector notation
    • A unit vector has a magnitude of 1
  • In 2-dimensions i represents movement in the horizontal direction (right/left) and j represents the movement in the vertical direction (up/down)
    • For example: The vector (-4i + 3j) would mean 4 units in the negative horizontal (x) direction (i.e., left) and 3 units in the positive vertical (y) direction (i.e., up)
  • In 3-dimensions i represents movement in the x direction (length), j represents movement in the y direction (width) and k represents the movement in the z direction (depth)
    • For example: The vector (-4i + 3j ­- k) would mean 4 units in the negative x direction, 3 units in the positive y direction and 1 unit in the negative z direction
  • As they are vectors, i, j and k are displayed in bold in textbooks and online but in handwriting they would be underlined (i, j and k)

3-9-1-ib-aa-hl-displacement-vectors-diagram-1

Exam Tip

  • Practice working with all types of vector notation so that you are prepared for whatever comes up in the exam
    • Your working and answer in the exam can be in any form unless told otherwise
    • It is generally best to write your final answer in the same form as given in the question, however you will not lose marks for not doing this unless it is specified in the question 
  • Vectors appear in bold (non-italic) font in textbooks and on exam papers, etc (i.e. F, α ) but in handwriting should be underlined (i.e. F , α )

Worked Example

a)
Write the vector begin mathsize 16px style open parentheses fraction numerator table row cell negative 4 end cell row 0 end table over denominator 5 end fraction close parentheses end style using base vector notation.

3-9-1-ib-aa-hl-vector-notation-we-solution-a-

b)

Write the vector bold k minus 2 bold j using column vector notation.

3-9-1-ib-aa-hl-vector-notation-we-solution-b-

Parallel Vectors

How do you know if two vectors are parallel?

  • Two vectors are parallel if one is a scalar multiple of the other
    • This means that all components of the vector have been multiplied by a common constant (scalar)
  • Multiplying every component in a vector by a scalar will change the magnitude of the vector but not the direction
    • For example: the vectors begin mathsize 16px style bold a equals blank open parentheses fraction numerator 1 over denominator table row 0 row 3 end table end fraction close parentheses end style and begin mathsize 16px style bold b equals 2 bold a equals blank 2 open parentheses fraction numerator 1 over denominator table row 0 row 3 end table end fraction close parentheses equals blank open parentheses fraction numerator 2 over denominator table row 0 row 6 end table end fraction close parentheses end style will have the same direction but the vector b will have twice the magnitude of a
      • They are parallel
  • If a vector can be factorised by a scalar then it is parallel to any scalar multiple of the factorised vector
    • For example: The vector 9i + 6j – 3k can be factorised by the scalar 3 to 3(3i + 2j – k) so the vector 9i + 6j – 3k is parallel to any scalar multiple of 3i + 2j – k
  • If a vector is multiplied by a negative scalar its direction will be reversed
    • It will still be parallel to the original vector
  • Two vectors are parallel if they have the same or reverse direction and equal if they have the same size and direction

Vector Addition Diagram 2

Exam Tip

  • It is easiest to spot that two vectors are parallel when they are in column vector notation
    • in your exam by writing vectors in column vector form and looking for a scalar multiple you will be able to quickly determine whether they are parallel or not

Worked Example

Show that the vectors bold a space equals space open parentheses fraction numerator table row 2 row 0 end table over denominator negative 4 end fraction close parentheses and bold b space equals space 6 bold k space – space 3 bold i are parallel and find the scalar multiple that maps a onto b.

3-9-2-ib-aa-hl-parallel-vectors-we-solution



  • 1. Number & Algebra
    • 1.1 Number & Algebra Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
          • 1.1.3 Partial Fractions
          • 1.2 Exponentials & Logs
            • 1.2.1 Introduction to Logarithms
              • 1.2.2 Laws of Logarithms
                • 1.2.3 Solving Exponential Equations
                • 1.3 Sequences & Series
                  • 1.3.1 Language of Sequences & Series
                    • 1.3.2 Arithmetic Sequences & Series
                      • 1.3.3 Geometric Sequences & Series
                        • 1.3.4 Applications of Sequences & Series
                          • 1.3.5 Compound Interest & Depreciation
                          • 1.4 Simple Proof & Reasoning
                            • 1.4.1 Proof
                            • 1.5 Further Proof & Reasoning
                              • 1.5.1 Proof by Induction
                                • 1.5.2 Proof by Contradiction
                                • 1.6 Binomial Theorem
                                  • 1.6.1 Binomial Theorem
                                    • 1.6.2 Extension of The Binomial Theorem
                                    • 1.7 Permutations & Combinations
                                      • 1.7.1 Counting Principles
                                        • 1.7.2 Permutations & Combinations
                                        • 1.8 Complex Numbers
                                          • 1.8.1 Intro to Complex Numbers
                                            • 1.8.2 Modulus & Argument
                                              • 1.8.3 Introduction to Argand Diagrams
                                              • 1.9 Further Complex Numbers
                                                • 1.9.1 Geometry of Complex Numbers
                                                  • 1.9.2 Forms of Complex Numbers
                                                    • 1.9.3 Complex Roots of Polynomials
                                                      • 1.9.4 De Moivre's Theorem
                                                        • 1.9.5 Roots of Complex Numbers
                                                        • 1.10 Systems of Linear Equations
                                                          • 1.10.1 Systems of Linear Equations
                                                            • 1.10.2 Algebraic Solutions
                                                          • 2. Functions
                                                            • 2.1 Linear Functions & Graphs
                                                              • 2.1.1 Equations of a Straight Line
                                                              • 2.2 Quadratic Functions & Graphs
                                                                • 2.2.1 Quadratic Functions
                                                                  • 2.2.2 Factorising & Completing the Square
                                                                    • 2.2.3 Solving Quadratics
                                                                      • 2.2.4 Quadratic Inequalities
                                                                        • 2.2.5 Discriminants
                                                                        • 2.3 Functions Toolkit
                                                                          • 2.3.1 Language of Functions
                                                                            • 2.3.2 Composite & Inverse Functions
                                                                              • 2.3.3 Symmetry of Functions
                                                                                • 2.3.4 Graphing Functions
                                                                                • 2.4 Other Functions & Graphs
                                                                                  • 2.4.1 Exponential & Logarithmic Functions
                                                                                    • 2.4.2 Solving Equations
                                                                                      • 2.4.3 Modelling with Functions
                                                                                      • 2.5 Reciprocal & Rational Functions
                                                                                        • 2.5.1 Reciprocal & Rational Functions
                                                                                        • 2.6 Transformations of Graphs
                                                                                          • 2.6.1 Translations of Graphs
                                                                                            • 2.6.2 Reflections of Graphs
                                                                                              • 2.6.3 Stretches Graphs
                                                                                                • 2.6.4 Composite Transformations of Graphs
                                                                                                • 2.7 Polynomial Functions
                                                                                                  • 2.7.1 Factor & Remainder Theorem
                                                                                                    • 2.7.2 Polynomial Division
                                                                                                      • 2.7.3 Polynomial Functions
                                                                                                        • 2.7.4 Roots of Polynomials
                                                                                                        • 2.8 Inequalities
                                                                                                          • 2.8.1 Solving Inequalities Graphically
                                                                                                            • 2.8.2 Polynomial Inequalities
                                                                                                            • 2.9 Further Functions & Graphs
                                                                                                              • 2.9.1 Modulus Functions
                                                                                                                • 2.9.2 Modulus Transformations
                                                                                                                  • 2.9.3 Modulus Equations & Inequalities
                                                                                                                    • 2.9.4 Reciprocal & Square Transformations
                                                                                                                  • 3. Geometry & Trigonometry
                                                                                                                    • 3.1 Geometry Toolkit
                                                                                                                      • 3.1.1 Coordinate Geometry
                                                                                                                        • 3.1.2 Radian Measure
                                                                                                                          • 3.1.3 Arcs & Sectors
                                                                                                                          • 3.2 Geometry of 3D Shapes
                                                                                                                            • 3.2.1 3D Coordinate Geometry
                                                                                                                              • 3.2.2 Volume & Surface Area
                                                                                                                              • 3.3 Trigonometry Toolkit
                                                                                                                                • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                                                  • 3.3.2 Non Right-Angled Trigonometry
                                                                                                                                    • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                                                    • 3.4 Trigonometry
                                                                                                                                      • 3.4.1 The Unit Circle
                                                                                                                                        • 3.4.2 Exact Values
                                                                                                                                        • 3.5 Trigonometric Functions & Graphs
                                                                                                                                          • 3.5.1 Graphs of Trigonometric Functions
                                                                                                                                            • 3.5.2 Transformations of Trigonometric Functions
                                                                                                                                              • 3.5.3 Modelling with Trigonometric Functions
                                                                                                                                              • 3.6 Trigonometric Equations & Identities
                                                                                                                                                • 3.6.1 Simple Identities
                                                                                                                                                  • 3.6.2 Compound Angle Formulae
                                                                                                                                                    • 3.6.3 Double Angle Formulae
                                                                                                                                                      • 3.6.4 Relationship Between Trigonometric Ratios
                                                                                                                                                        • 3.6.5 Linear Trigonometric Equations
                                                                                                                                                          • 3.6.6 Quadratic Trigonometric Equations
                                                                                                                                                          • 3.7 Inverse & Reciprocal Trig Functions
                                                                                                                                                            • 3.7.1 Reciprocal Trig Functions
                                                                                                                                                              • 3.7.2 Inverse Trig Functions
                                                                                                                                                              • 3.8 Further Trigonometry
                                                                                                                                                                • 3.8.1 Trigonometric Proof
                                                                                                                                                                  • 3.8.2 Strategy for Trigonometric Equations
                                                                                                                                                                  • 3.9 Vector Properties
                                                                                                                                                                    • 3.9.1 Introduction to Vectors
                                                                                                                                                                      • 3.9.2 Position & Displacement Vectors
                                                                                                                                                                        • 3.9.3 Magnitude of a Vector
                                                                                                                                                                          • 3.9.4 The Scalar Product
                                                                                                                                                                            • 3.9.5 Geometric Proof with Vectors
                                                                                                                                                                            • 3.10 Vector Equations of Lines
                                                                                                                                                                              • 3.10.1 Vector Equations of Lines
                                                                                                                                                                                • 3.10.2 Applications to Kinematics
                                                                                                                                                                                  • 3.10.3 Pairs of Lines in 3D
                                                                                                                                                                                    • 3.10.4 The Vector Product
                                                                                                                                                                                      • 3.10.5 Shortest Distances with Lines
                                                                                                                                                                                      • 3.11 Vector Planes
                                                                                                                                                                                        • 3.11.1 Vector Equations of Planes
                                                                                                                                                                                          • 3.11.2 Intersections of Lines & Planes
                                                                                                                                                                                            • 3.11.3 Angles Between Lines & Planes
                                                                                                                                                                                              • 3.11.4 Shortest Distances with Planes
                                                                                                                                                                                            • 4. Statistics & Probability
                                                                                                                                                                                              • 4.1 Statistics Toolkit
                                                                                                                                                                                                • 4.1.1 Sampling & Data Collection
                                                                                                                                                                                                  • 4.1.2 Statistical Measures
                                                                                                                                                                                                    • 4.1.3 Frequency Tables
                                                                                                                                                                                                      • 4.1.4 Linear Transformations of Data
                                                                                                                                                                                                        • 4.1.5 Outliers
                                                                                                                                                                                                          • 4.1.6 Univariate Data
                                                                                                                                                                                                            • 4.1.7 Interpreting Data
                                                                                                                                                                                                            • 4.2 Correlation & Regression
                                                                                                                                                                                                              • 4.2.1 Bivariate Data
                                                                                                                                                                                                                • 4.2.2 Correlation & Regression
                                                                                                                                                                                                                • 4.3 Probability
                                                                                                                                                                                                                  • 4.3.1 Probability & Types of Events
                                                                                                                                                                                                                    • 4.3.2 Conditional Probability
                                                                                                                                                                                                                      • 4.3.3 Bayes' Theorem
                                                                                                                                                                                                                        • 4.3.4 Sample Space Diagrams
                                                                                                                                                                                                                        • 4.4 Probability Distributions
                                                                                                                                                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                                                                                                                                                            • 4.4.2 Mean & Variance
                                                                                                                                                                                                                            • 4.5 Binomial Distribution
                                                                                                                                                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                                                                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                                                                                                                • 4.6 Normal Distribution
                                                                                                                                                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                                                                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                                                                                                                                                      • 4.6.3 Standardisation of Normal Variables
                                                                                                                                                                                                                                      • 4.7 Further Probability Distributions
                                                                                                                                                                                                                                        • 4.7.1 Probability Density Function
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                  • 5.2.3 Higher Order Derivatives
                                                                                                                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                                                                                                                        • 5.3 Integration
                                                                                                                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                          • 5.7 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.7.1 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.8 Advanced Differentiation
                                                                                                                                                                                                                                                                              • 5.8.1 First Principles Differentiation
                                                                                                                                                                                                                                                                                • 5.8.2 Applications of Chain Rule
                                                                                                                                                                                                                                                                                  • 5.8.3 Implicit Differentiation
                                                                                                                                                                                                                                                                                    • 5.8.4 Differentiating Further Functions
                                                                                                                                                                                                                                                                                    • 5.9 Advanced Integration
                                                                                                                                                                                                                                                                                      • 5.9.1 Integrating Further Functions
                                                                                                                                                                                                                                                                                        • 5.9.2 Further Techniques of Integration
                                                                                                                                                                                                                                                                                          • 5.9.3 Integrating with Partial Fractions
                                                                                                                                                                                                                                                                                            • 5.9.4 Advanced Applications of Integration
                                                                                                                                                                                                                                                                                              • 5.9.5 Modelling with Volumes of Revolution
                                                                                                                                                                                                                                                                                              • 5.10 Differential Equations
                                                                                                                                                                                                                                                                                                • 5.10.1 Numerical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                  • 5.10.2 Analytical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.10.3 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.11 MacLaurin Series
                                                                                                                                                                                                                                                                                                      • 5.11.1 Maclaurin Series
                                                                                                                                                                                                                                                                                                        • 5.11.2 Maclaurin Series from Differential Equations
                                                                                                                                                                                                                                                                                                        • 5.12 Further Limits (inc l'Hôpital's Rule)
                                                                                                                                                                                                                                                                                                          • 5.12.1 Further Limits
                                                                                                                                                                                                                                                                                                        Amber Perry

                                                                                                                                                                                                                                                                                                        Author: Amber

                                                                                                                                                                                                                                                                                                        Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.


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