Vectors can represent both positions and movements. Positions use coordinates (x,y) while movements use vectors with the change in x and y written vertically. Vectors can be added by adding the x and y components. Vectors can also be scaled by multiplying each component by a scalar value. Exam questions often involve finding a vector in part (a) and then using it to find another vector in part (b).
This document consists of a 16 page exam for the Cambridge International Examinations General Certificate of Education Advanced Subsidiary Level and Advanced Level Biology exam. It contains 40 multiple choice questions testing various concepts in biology such as cell structure, transport processes, enzymes, and ecology.
This document contains a collection of questions from various topics including mathematics, communication, electronics, and GEAS (presumably general engineering and applied science) along with their answers. The questions range from easy to difficult and cover concepts such as antenna gain, probability, stress and strain in materials, signal modulation, and more.
Here are the steps to solve this problem:
a) OX = λ(a + b) = λa + λb
b) BX = μBQ = μ(b - a/3) = μb - μa/3
OX = OB - BX = b - (μb - μa/3) = (1 - μ)b + μa/3
c) Comparing the coefficients of a and b in the two expressions for OX obtained in parts a) and b) gives:
λ = μ/3 and 1 - μ = λ
Solving these simultaneously gives λ = μ = 1/3
Therefore, the ratio O
This document provides an overview of vectors in 3 dimensions. It discusses finding the distance between points using Pythagoras' theorem, writing vectors in i, j, k notation, finding the magnitude of a vector, and finding the angle between a vector and the coordinate axes. It also provides examples of using vectors to solve geometric problems by drawing diagrams and comparing coefficients of vectors. The exercises involve finding distances, directions of vectors, parallel vectors, and using vector operations to find missing points to form geometric shapes like parallelograms.
1. The document discusses integration and areas under curves. It provides examples of indefinite integration by finding antiderivatives given derivative functions.
2. Definite integration allows finding the exact area under a curve between limits, unlike indefinite integration which has a constant of integration. Examples are worked through of evaluating definite integrals.
3. Problem solving with definite integrals is demonstrated, such as finding the possible values of a constant P given the value of a definite integral involving P.
This document provides an overview of unit 2 on the algebra of vectors from the course EMA 310: Vectors and Mechanics. It introduces the learning objectives which are to find the resultant of given vectors, add vectors using the parallelogram and triangle laws of addition, and establish and use properties of vector addition. Examples are given of applying the triangle and parallelogram laws of vector addition, along with activities for students to practice finding vector sums and multiplying vectors by scalars.
1. This document discusses graph transformations of functions, including translations, stretches, and reflections. It provides rules for how modifications inside and outside the function f(x) will affect the x-values and y-values of the graph.
2. Examples are given of applying transformation rules to specific points on a graph and determining the new coordinates. The document also demonstrates sketching a transformed graph using key points.
3. An exercise section provides multiple choice and short answer questions to test understanding of describing transformations and finding coordinates of transformed points.
This document consists of a 16 page exam for the Cambridge International Examinations General Certificate of Education Advanced Subsidiary Level and Advanced Level Biology exam. It contains 40 multiple choice questions testing various concepts in biology such as cell structure, transport processes, enzymes, and ecology.
This document contains a collection of questions from various topics including mathematics, communication, electronics, and GEAS (presumably general engineering and applied science) along with their answers. The questions range from easy to difficult and cover concepts such as antenna gain, probability, stress and strain in materials, signal modulation, and more.
Here are the steps to solve this problem:
a) OX = λ(a + b) = λa + λb
b) BX = μBQ = μ(b - a/3) = μb - μa/3
OX = OB - BX = b - (μb - μa/3) = (1 - μ)b + μa/3
c) Comparing the coefficients of a and b in the two expressions for OX obtained in parts a) and b) gives:
λ = μ/3 and 1 - μ = λ
Solving these simultaneously gives λ = μ = 1/3
Therefore, the ratio O
This document provides an overview of vectors in 3 dimensions. It discusses finding the distance between points using Pythagoras' theorem, writing vectors in i, j, k notation, finding the magnitude of a vector, and finding the angle between a vector and the coordinate axes. It also provides examples of using vectors to solve geometric problems by drawing diagrams and comparing coefficients of vectors. The exercises involve finding distances, directions of vectors, parallel vectors, and using vector operations to find missing points to form geometric shapes like parallelograms.
1. The document discusses integration and areas under curves. It provides examples of indefinite integration by finding antiderivatives given derivative functions.
2. Definite integration allows finding the exact area under a curve between limits, unlike indefinite integration which has a constant of integration. Examples are worked through of evaluating definite integrals.
3. Problem solving with definite integrals is demonstrated, such as finding the possible values of a constant P given the value of a definite integral involving P.
This document provides an overview of unit 2 on the algebra of vectors from the course EMA 310: Vectors and Mechanics. It introduces the learning objectives which are to find the resultant of given vectors, add vectors using the parallelogram and triangle laws of addition, and establish and use properties of vector addition. Examples are given of applying the triangle and parallelogram laws of vector addition, along with activities for students to practice finding vector sums and multiplying vectors by scalars.
1. This document discusses graph transformations of functions, including translations, stretches, and reflections. It provides rules for how modifications inside and outside the function f(x) will affect the x-values and y-values of the graph.
2. Examples are given of applying transformation rules to specific points on a graph and determining the new coordinates. The document also demonstrates sketching a transformed graph using key points.
3. An exercise section provides multiple choice and short answer questions to test understanding of describing transformations and finding coordinates of transformed points.
Linear equations in two variables. Please download the powerpoint file to enable animation.
Disclaimer: Some parts of the presentation are obtained from various sources. Credit to the rightful owners.
This document discusses vector algebra concepts including:
1. Vectors can represent quantities that have both magnitude and direction, unlike scalars which only have magnitude.
2. Common vector operations include addition, subtraction, and determining the resultant or sum of multiple vectors.
3. The dot product of two vectors produces a scalar value that can indicate whether vectors are parallel or perpendicular and define physical quantities like work and electric fields.
4. The cross product of two vectors produces a new vector that is perpendicular to the original vectors and can define quantities like angular velocity and motion in electromagnetic fields.
The document discusses key concepts for straight line equations in GCSE mathematics including:
- Understanding that equations of the form y=mx+c correspond to straight line graphs
- Plotting graphs from their equations and finding gradients and intercepts
- Relating gradients to parallel and perpendicular lines
- Generating equations for lines parallel or perpendicular to given lines
- Finding gradients and equations from two points on a line
This document introduces vectors and how they can be used to describe displacements and solve problems involving displacement. It discusses that vectors have both direction and magnitude, and provides examples. Vectors can be added and represented using line segments. The triangle law of addition allows vectors to be added using a triangle. Vectors can also be described using i and j notation, where i and j are unit vectors along the x and y axes, and any two-dimensional vector can be written as ai + bj. Problems can then be solved by adding or subtracting the i and j terms. The magnitude of a vector can be found using Pythagoras' theorem, and the angle between a vector and an axis can be found using trig
X matematika peminatan kd 3.2_final (2)DeraAnnisa1
1) The document discusses vectors in three-dimensional space (R3), including calculating the modulus (length) of vectors, determining vector position, and representing vectors using Cartesian coordinates, column vectors, and linear combinations of basis vectors.
2) It provides examples of finding equations and representations of vectors between points on geometric shapes in R3.
3) The document defines the vector position of a point P in Cartesian coordinates as the vector OP from the origin O to point P, with the coordinates of P being the components of the position vector OP.
1. The document discusses vector calculus concepts including the dot product, cross product, gradient, divergence, and curl of vectors. It provides examples and explanations of how to calculate these quantities.
2. The dot product yields a scalar quantity and is used to calculate the angle between two vectors. The cross product yields a vector that is perpendicular to the plane of the two input vectors.
3. The gradient is the vector of partial derivatives of a scalar function and points in the direction of maximum increase. The divergence and curl are used to describe vector fields and involve taking partial derivatives.
This document provides an introduction to rational numbers. It defines rational numbers as numbers that can be written as fractions where the numerator and denominator are integers. This includes integers. It explains that rational numbers can be written in different equivalent forms by multiplying or dividing the numerator and denominator by the same integer. The document then discusses how to add, subtract, multiply and divide rational numbers. It presents the algebraic rules for doing each operation and provides examples. It also explains how to check if two fractions are equal by seeing if their numerators times denominators are the same.
This document provides an introduction to rational numbers. It defines rational numbers as numbers that can be written as fractions where the numerator and denominator are integers. This includes integers. It explains that rational numbers can be written in different equivalent forms by multiplying or dividing the numerator and denominator by the same integer. The document then discusses how to add, subtract, multiply and divide rational numbers. It presents the algebraic rules for doing each operation and provides examples. It also explains how to check if two fractions are equal by seeing if their numerators times the other's denominator equals the other's numerator times the denominator.
This document provides an introduction to rational numbers. It defines rational numbers as numbers that can be written as fractions where the numerator and denominator are integers. This includes integers. It explains that rational numbers can be written in different equivalent forms by multiplying or dividing the numerator and denominator by the same integer. The document then discusses how to add, subtract, multiply and divide rational numbers. It presents the algebraic rules for doing each operation and provides examples. It also explains how to check if two fractions are equal by seeing if their numerators times the other's denominator equals the other's numerator times the denominator.
The document defines a quadratic equation as a second-degree equation in one variable that can be written in the form ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. It provides examples of quadratic and non-quadratic equations, and shows how to identify the coefficients a, b, and c when a quadratic equation is written in standard form. Activities are included for students to practice sorting quadratic equations and rewriting them in standard form.
This document discusses differentiation and finding the gradient function or derivative of functions. It begins by explaining how to find the derivative of a function numerically by approximating the gradient between two nearby points and then taking the limit as the distance between the points approaches zero. This leads to a general formula for finding the derivative using limits. Examples are then provided of finding the derivatives of polynomial functions like x^2 and x^4 using this first principles approach. The document then introduces a quicker rule for differentiating terms of the form x^n, where the derivative is nx^{n-1}. Several examples are given of applying this rule.
Lesson plan in mathematics 9 (illustrations of quadratic equations)Decena15
The lesson plan outlines a lesson on quadratic equations. It introduces quadratic equations and their standard form of ax2 + bx + c = 0. Examples are provided to illustrate how to write quadratic equations in standard form given values of a, b, and c or when expanding multiplied linear expressions. Students complete an activity identifying linear and quadratic equations. They are then assessed by writing equations in standard form and identifying the values of a, b, and c.
Functions ppt Dr Frost Maths Mixed questionsgcutbill
The document provides an overview of functions topics for GCSE/IGCSE mathematics, including understanding functions, inverse functions, composite functions, domain and range of functions, and piecewise functions. It contains examples of different types of functions and exercises for students to practice evaluating functions, finding inverse functions, and solving word problems involving functions. The document is intended to help students learn and teachers teach key concepts related to functions.
The document discusses solving equations, including equations with unknowns on both sides and with brackets. It provides examples of solving various types of equations, such as equations with fractions or variables on both sides. Strategies for solving equations include collecting like terms, using the inverse operation to isolate the variable, and expanding any brackets before solving.
The document discusses differentiation, explaining how to find the gradient function or derivative of a curve using limits, and providing examples of finding the derivative of polynomials like x^2, x^4, and x^n using a general rule of multiplying the term by its power and reducing the power by 1. It also discusses notation for derivatives and introduces the concept of using differentiation to find equations of tangents and analyze functions.
The document discusses differentiation, explaining how to find the gradient function or derivative of a curve using limits, and providing examples of finding the derivative of polynomials like x^2, x^4, and x^n using differentiation rules rather than evaluating limits directly. It also covers finding the equations of tangents, determining increasing and decreasing functions, and identifying stationary points.
The document discusses linear equations in two variables. It will cover writing linear equations in standard and slope-intercept form, graphing linear equations using two points, intercepts and slope/point, and describing graphs by their intercepts and slope. Key topics include defining the standard form as Ax + By = C, rewriting equations between the two forms, using two points, x-intercept, y-intercept or slope/point to graph, and describing graphs by their slope and intercepts.
This document contains 17 multi-part math problems related to straight lines and graphs. The problems cover topics such as finding the gradient and equation of a line, determining if two lines are parallel or perpendicular, finding the midpoint and equation of a line through two given points, and interpreting the gradient of a graph. Several questions provide diagrams or graphs to accompany the problems. The document also provides attribution information, noting the exam boards and sample assessment materials that the questions were retrieved from.
The document discusses key concepts in vectors including:
- Vectors can be represented geometrically as arrows or algebraically as ordered lists of numbers in a coordinate system.
- The two fundamental vector operations are vector addition and scalar multiplication. Vector addition involves combining the components of vectors, while scalar multiplication scales the magnitude and direction of a vector.
- Basis vectors define a coordinate system. Any vector can be written as a linear combination of basis vectors using scalar multiplication and vector addition. In 2D, the standard basis vectors are i and j along the x- and y-axes.
- The linear span of vectors is the set of all possible linear combinations of those vectors. If the vectors are linearly independent
The document discusses power series representations of functions. Power series allow functions that cannot be integrated, like ex2, to be approximated by polynomials which can be integrated. This allows integrals to be approximated to any degree of accuracy. Power series also allow irrational numbers like π to be expressed as an infinite decimal expansion. Power series approximations can be used to find approximate solutions to difficult differential equations.
Linear equations in two variables. Please download the powerpoint file to enable animation.
Disclaimer: Some parts of the presentation are obtained from various sources. Credit to the rightful owners.
This document discusses vector algebra concepts including:
1. Vectors can represent quantities that have both magnitude and direction, unlike scalars which only have magnitude.
2. Common vector operations include addition, subtraction, and determining the resultant or sum of multiple vectors.
3. The dot product of two vectors produces a scalar value that can indicate whether vectors are parallel or perpendicular and define physical quantities like work and electric fields.
4. The cross product of two vectors produces a new vector that is perpendicular to the original vectors and can define quantities like angular velocity and motion in electromagnetic fields.
The document discusses key concepts for straight line equations in GCSE mathematics including:
- Understanding that equations of the form y=mx+c correspond to straight line graphs
- Plotting graphs from their equations and finding gradients and intercepts
- Relating gradients to parallel and perpendicular lines
- Generating equations for lines parallel or perpendicular to given lines
- Finding gradients and equations from two points on a line
This document introduces vectors and how they can be used to describe displacements and solve problems involving displacement. It discusses that vectors have both direction and magnitude, and provides examples. Vectors can be added and represented using line segments. The triangle law of addition allows vectors to be added using a triangle. Vectors can also be described using i and j notation, where i and j are unit vectors along the x and y axes, and any two-dimensional vector can be written as ai + bj. Problems can then be solved by adding or subtracting the i and j terms. The magnitude of a vector can be found using Pythagoras' theorem, and the angle between a vector and an axis can be found using trig
X matematika peminatan kd 3.2_final (2)DeraAnnisa1
1) The document discusses vectors in three-dimensional space (R3), including calculating the modulus (length) of vectors, determining vector position, and representing vectors using Cartesian coordinates, column vectors, and linear combinations of basis vectors.
2) It provides examples of finding equations and representations of vectors between points on geometric shapes in R3.
3) The document defines the vector position of a point P in Cartesian coordinates as the vector OP from the origin O to point P, with the coordinates of P being the components of the position vector OP.
1. The document discusses vector calculus concepts including the dot product, cross product, gradient, divergence, and curl of vectors. It provides examples and explanations of how to calculate these quantities.
2. The dot product yields a scalar quantity and is used to calculate the angle between two vectors. The cross product yields a vector that is perpendicular to the plane of the two input vectors.
3. The gradient is the vector of partial derivatives of a scalar function and points in the direction of maximum increase. The divergence and curl are used to describe vector fields and involve taking partial derivatives.
This document provides an introduction to rational numbers. It defines rational numbers as numbers that can be written as fractions where the numerator and denominator are integers. This includes integers. It explains that rational numbers can be written in different equivalent forms by multiplying or dividing the numerator and denominator by the same integer. The document then discusses how to add, subtract, multiply and divide rational numbers. It presents the algebraic rules for doing each operation and provides examples. It also explains how to check if two fractions are equal by seeing if their numerators times denominators are the same.
This document provides an introduction to rational numbers. It defines rational numbers as numbers that can be written as fractions where the numerator and denominator are integers. This includes integers. It explains that rational numbers can be written in different equivalent forms by multiplying or dividing the numerator and denominator by the same integer. The document then discusses how to add, subtract, multiply and divide rational numbers. It presents the algebraic rules for doing each operation and provides examples. It also explains how to check if two fractions are equal by seeing if their numerators times the other's denominator equals the other's numerator times the denominator.
This document provides an introduction to rational numbers. It defines rational numbers as numbers that can be written as fractions where the numerator and denominator are integers. This includes integers. It explains that rational numbers can be written in different equivalent forms by multiplying or dividing the numerator and denominator by the same integer. The document then discusses how to add, subtract, multiply and divide rational numbers. It presents the algebraic rules for doing each operation and provides examples. It also explains how to check if two fractions are equal by seeing if their numerators times the other's denominator equals the other's numerator times the denominator.
The document defines a quadratic equation as a second-degree equation in one variable that can be written in the form ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. It provides examples of quadratic and non-quadratic equations, and shows how to identify the coefficients a, b, and c when a quadratic equation is written in standard form. Activities are included for students to practice sorting quadratic equations and rewriting them in standard form.
This document discusses differentiation and finding the gradient function or derivative of functions. It begins by explaining how to find the derivative of a function numerically by approximating the gradient between two nearby points and then taking the limit as the distance between the points approaches zero. This leads to a general formula for finding the derivative using limits. Examples are then provided of finding the derivatives of polynomial functions like x^2 and x^4 using this first principles approach. The document then introduces a quicker rule for differentiating terms of the form x^n, where the derivative is nx^{n-1}. Several examples are given of applying this rule.
Lesson plan in mathematics 9 (illustrations of quadratic equations)Decena15
The lesson plan outlines a lesson on quadratic equations. It introduces quadratic equations and their standard form of ax2 + bx + c = 0. Examples are provided to illustrate how to write quadratic equations in standard form given values of a, b, and c or when expanding multiplied linear expressions. Students complete an activity identifying linear and quadratic equations. They are then assessed by writing equations in standard form and identifying the values of a, b, and c.
Functions ppt Dr Frost Maths Mixed questionsgcutbill
The document provides an overview of functions topics for GCSE/IGCSE mathematics, including understanding functions, inverse functions, composite functions, domain and range of functions, and piecewise functions. It contains examples of different types of functions and exercises for students to practice evaluating functions, finding inverse functions, and solving word problems involving functions. The document is intended to help students learn and teachers teach key concepts related to functions.
The document discusses solving equations, including equations with unknowns on both sides and with brackets. It provides examples of solving various types of equations, such as equations with fractions or variables on both sides. Strategies for solving equations include collecting like terms, using the inverse operation to isolate the variable, and expanding any brackets before solving.
The document discusses differentiation, explaining how to find the gradient function or derivative of a curve using limits, and providing examples of finding the derivative of polynomials like x^2, x^4, and x^n using a general rule of multiplying the term by its power and reducing the power by 1. It also discusses notation for derivatives and introduces the concept of using differentiation to find equations of tangents and analyze functions.
The document discusses differentiation, explaining how to find the gradient function or derivative of a curve using limits, and providing examples of finding the derivative of polynomials like x^2, x^4, and x^n using differentiation rules rather than evaluating limits directly. It also covers finding the equations of tangents, determining increasing and decreasing functions, and identifying stationary points.
The document discusses linear equations in two variables. It will cover writing linear equations in standard and slope-intercept form, graphing linear equations using two points, intercepts and slope/point, and describing graphs by their intercepts and slope. Key topics include defining the standard form as Ax + By = C, rewriting equations between the two forms, using two points, x-intercept, y-intercept or slope/point to graph, and describing graphs by their slope and intercepts.
This document contains 17 multi-part math problems related to straight lines and graphs. The problems cover topics such as finding the gradient and equation of a line, determining if two lines are parallel or perpendicular, finding the midpoint and equation of a line through two given points, and interpreting the gradient of a graph. Several questions provide diagrams or graphs to accompany the problems. The document also provides attribution information, noting the exam boards and sample assessment materials that the questions were retrieved from.
The document discusses key concepts in vectors including:
- Vectors can be represented geometrically as arrows or algebraically as ordered lists of numbers in a coordinate system.
- The two fundamental vector operations are vector addition and scalar multiplication. Vector addition involves combining the components of vectors, while scalar multiplication scales the magnitude and direction of a vector.
- Basis vectors define a coordinate system. Any vector can be written as a linear combination of basis vectors using scalar multiplication and vector addition. In 2D, the standard basis vectors are i and j along the x- and y-axes.
- The linear span of vectors is the set of all possible linear combinations of those vectors. If the vectors are linearly independent
The document discusses power series representations of functions. Power series allow functions that cannot be integrated, like ex2, to be approximated by polynomials which can be integrated. This allows integrals to be approximated to any degree of accuracy. Power series also allow irrational numbers like π to be expressed as an infinite decimal expansion. Power series approximations can be used to find approximate solutions to difficult differential equations.
Khushi Saini, An Intern from The Sparks Foundationkhushisaini0924
This is my first task as an Talent Acquisition(Human resources) Intern in The Sparks Foundation on Recruitment, article and posts.
I invitr everyone to look into my work and provide me a quick feedback.
5 key differences between Hard skill and Soft skillsRuchiRathor2
𝐓𝐡𝐞 𝐏𝐞𝐫𝐟𝐞𝐜𝐭 𝐁𝐥𝐞𝐧𝐝:
𝐖𝐡𝐲 𝐘𝐨𝐮 𝐍𝐞𝐞𝐝 𝐁𝐨𝐭𝐡 𝐇𝐚𝐫𝐝 & 𝐒𝐨𝐟𝐭 𝐒𝐤𝐢𝐥𝐥𝐬 𝐭𝐨 𝐓𝐡𝐫𝐢𝐯𝐞 💯
In today's dynamic and competitive market, a well-rounded skillset is no longer a luxury - it's a necessity.
While technical expertise (hard skills) is crucial for getting your foot in the door, it's the combination of hard and soft skills that propels you towards long-term success and career advancement. ✨
Think of it like this: Imagine a highly skilled carpenter with a masterful understanding of woodworking (hard skills). But if they struggle to communicate effectively with clients, collaborate with builders, or adapt to project changes (soft skills), their true potential remains untapped. 😐
The synergy between hard and soft skills is what creates true value in the workplace. Strong communication allows you to clearly articulate your technical expertise, while problem-solving skills help you navigate complex challenges alongside your team. 💫
By actively developing both sets of skills, you position yourself as a well-rounded professional who can not only perform tasks efficiently but also contribute meaningfully to a collaborative and dynamic work environment.
Go through the carousel and let me know your views 🤩
Learnings from Successful Jobs SearchersBruce Bennett
Are you interested to know what actions help in a job search? This webinar is the summary of several individuals who discussed their job search journey for others to follow. You will learn there are common actions that helped them succeed in their quest for gainful employment.
I am an accomplished and driven administrative management professional with a proven track record of supporting senior executives and managing administrative teams. I am skilled in strategic planning, project management, and organizational development, and have extensive experience in improving processes, enhancing productivity, and implementing solutions to support business objectives and growth.
LinkedIn for Your Job Search June 17, 2024Bruce Bennett
This webinar helps you understand and navigate your way through LinkedIn. Topics covered include learning the many elements of your profile, populating your work experience history, and understanding why a profile is more than just a resume. You will be able to identify the different features available on LinkedIn and where to focus your attention. We will teach how to create a job search agent on LinkedIn and explore job applications on LinkedIn.
LinkedIn Strategic Guidelines for June 2024Bruce Bennett
LinkedIn is a powerful tool for networking, researching, and marketing yourself to clients and employers. This session teaches strategic practices for building your LinkedIn internet presence and marketing yourself. The use of # and @ symbols is covered as well as going mobile with the LinkedIn app.
1. GCSE: Vectors
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
www.drfrostmaths.com
Last modified: 20th March 2017
2. 𝑥
𝑦
1
2
3
4
5
6
-1
-2
1 2 3 4 5 6 7
2,3
4
1
Coordinates represent a position.
We write 𝑥, 𝑦 to indicate the 𝑥
position and 𝑦 position.
Vectors represent a movement.
We write
𝑥
𝑦 to indicate the
change in 𝑥 and change in 𝑦.
?
?
?
?
Note we put the numbers
vertically. Do NOT write
4
1
;
there is no line and vectors are
very different to fractions!
You may have seen vectors
briefly before if you’ve done
transformations; we can use
them to describe the
movement of a shape in a
translation.
Coordinates vs Vectors
4. You’re used to variables representing numbers in maths. They can also
represent vectors!
𝒂
𝒃
𝑋
𝑌 What can you say about how we
use variables for vertices (points) vs
variables for vectors?
We use capital letters for vertices
and lower case letters for vectors.
There’s 3 ways in which can
represent the vector from point 𝑋
to 𝑍:
1. 𝒂 (in bold)
2. 𝑎 (with an ‘underbar’)
3. 𝑋𝑍
𝑍 ?
?
?
?
Writing Vectors
5. 𝒂
𝒃
𝑋
𝑌
𝑍
𝑋𝑍 = 𝒂 =
2
5
𝑍𝑌 = 𝒃 =
5
3
𝑋𝑌 =
7
8
?
?
?
What do you notice about the numbers in
7
8
when compared to
2
5
and
5
3
?
We’ve simply added the 𝑥 values and 𝑦
values to describe the combined movement.
i.e.
𝒂 + 𝒃 =
2
5
+
5
3
=
7
8
𝑋𝑍 + 𝑍𝑌 = 𝑋𝑌
?
Bro Important Note: The point is that we can use
any route to get from the start to finish, and the
vector will always be the same.
• Route 1: We go from 𝑋 to 𝑌 via 𝑍.
𝑋𝑍 =
2
5
+
5
3
=
7
8
• Route 2: Use the direct line from 𝑋 to 𝑌:
7
8
Adding Vectors
6. 𝒂
Scaling Vectors
We can ‘scale’ a vector by
multiplying it by a normal number,
aptly known as a scalar.
If 𝒂 =
4
3
, then
2𝒂 = 2
4
3
=
8
6
What is the same about 𝒂 and 2𝒂 and
what is different?
2𝒂
Bro Note: Note that
vector letters are bold
but scalars are not.
Same:
Same direction / Parallel
Different:
The length of the vector, known as
the magnitude, is longer.
?
?
?
7. More on Adding/Subtracting Vectors
𝐴
𝑋
𝒂
𝒃
2𝒄
If 𝑂𝐴 = 𝒂, 𝐴𝐵 = 𝒃 and 𝑋𝐵 = 2𝒄,
then find the following in terms
of 𝑎, 𝑏 and 𝑐:
𝑂𝐵 = 𝒂 + 𝒃
𝑂𝑌 = 𝒂 + 2𝒃
𝐴𝑋 = 𝒃 − 2𝒄
𝑋𝑂 = 2𝒄 − 𝒃 − 𝒂
𝑌𝑋 = −𝒃 − 2𝒄
Note: Since −
𝑥
𝑦 =
−𝑥
−𝑦 ,
subtracting a vector goes in
the opposite direction.
?
?
?
𝒃
𝐵
𝑌
𝑂 𝒃
𝒂
Bro Side Note: Since 𝑏 + 𝑎
would end up at the same
finish point, we can see
𝒃 + 𝒂 = 𝒂 + 𝒃 (i.e. vector
addition, like normal
addition, is ‘commutative’)
?
?
8. a. 𝐵𝐴 = −𝑎
b. 𝐴𝐶 = 𝑎 + 𝑏
c. 𝐷𝐵 = 2𝑎 − 𝑏
d. 𝐴𝐷 = −𝑎 + 𝑏
a. 𝑀𝐾 = −𝑎 − 𝑏
b. 𝑁𝐿 = 3𝑎 − 𝑏
c. 𝑁𝐾 = 2𝑎 − 𝑏
d. 𝐾𝑁 = −2𝑎 + 𝑏
a. 𝑍𝑋 = 𝑎 + 𝑏
b. 𝑌𝑊 = 𝑎 − 2𝑏
c. 𝑋𝑌 = −𝑎 + 𝑏
d. 𝑋𝑍 = −𝑎 − 𝑏
a. 𝐴𝐵 = −𝑎 + 𝑏
b. 𝐹𝑂 = −𝑎 + 𝑏
c. 𝐴𝑂 = −2𝑎 + 𝑏
d. 𝐹𝐷 = −3𝑎 + 2𝑏
2
3
4
5
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Exercise 1 (on provided sheet)
1 State the value of each vector.
𝒂
𝒃
𝒄
𝒅
𝒆 𝒇
𝒂 =
3
1
𝒃 =
3
−2
𝒄 =
−3
0
𝒅 =
0
5
𝒆 =
−1
4
𝒇 =
−3
−4
? ? ?
? ? ?
10. The ‘Two Parter’ exam question
Many exams questions follow a two-part format:
a) Find a relatively easy vector using skills from Lesson 1.
b) Find a harder vector that uses a fraction of your vector from part (a).
Edexcel June 2013 1H Q27
𝑆𝑄 = −𝒃 + 𝒂
For (b), there’s two possible paths
to get from 𝑁 to 𝑅: via 𝑆 or via 𝑄.
But which is best?
In (a) we found 𝑺 to 𝑸 rather than
𝑸 to 𝑺, so it makes sense to go in
this direction so that we can use
our result in (a).
a
b 𝑁𝑅 =
2
5
𝑆𝑄 + 𝒃
=
2
5
−𝒃 + 𝒂 + 𝒃
=
2
5
𝒂 +
3
5
𝒃
Bro Tip: This ratio
wasn’t in the
original diagram. I
like to add the ratio
as a visual aid.
?
?
Bro Workings Tip: While you’re
welcome to start your working with
the second line, I recommend the
first line so that your chosen route
is clearer.
𝑄𝑅 is also 𝑏
because it is exactly
the same
movement as 𝑃𝑆.
?
11. Test Your Understanding
𝐴𝐵 = 𝐴𝑂 + 𝑂𝐵
= −𝒂 + 𝒃
?
𝑂𝑃 = 𝒂 +
3
4
𝐴𝐵
= 𝒂 +
3
4
−𝒂 + 𝒃
=
1
4
𝒂 +
3
4
𝒃
?
You MUST
expand and
simplify.
Edexcel June 2012
17. Types of vector ‘proof’ questions
“Prove that … is
a straight line.”
“Show that … is
parallel to …”
“Prove these two
vectors are equal.”
18. ?
Showing vectors are equal
There is nothing
mysterious about this
kind of ‘prove’ question.
Just find any vectors
involved, in this case,
𝑂𝑋 and
2
5
𝑂𝑌. Hopefully
they should be equal!
Edexcel March 2013
2
5
𝑂𝑌 =
2
5
6𝑏 + 5𝑎 − 𝑏
= 2𝑎 + 2𝑏
𝑂𝑋 = 3𝑎 +
1
3
−3𝑎 + 6𝑏
= 2𝑎 + 2𝑏
∴ 𝑂𝑋 =
2
5
𝑂𝑌
With proof questions
you should restate the
thing you are trying to
prove, as a ‘conclusion’.
?
19. What do you notice?
𝒂 2𝒂 𝒂 + 2𝒃
2𝒂 + 4𝒃
𝒂
−
3
2
𝒂
! Vectors are parallel
if one is a multiple of
another.
?
20. Parallel or not parallel?
Vector 1 Vector 2 Parallel?
(in general)
𝑎 + 𝑏 𝑎 + 2𝑏
𝑎 + 𝑏 3𝑎 + 3𝑏
𝑎 + 2𝑏 −3𝑎 − 6𝑏
𝑎 − 𝑏 −𝑎 + 𝑏
No Yes
No Yes
No Yes
No Yes
For ones which are parallel, show it diagrammatically.
21. How to show two vectors are parallel
𝒂
𝑴
𝑶 𝑩
𝑿
𝒃
𝑨 𝑪
𝑋 is a point on 𝐴𝐵 such that 𝐴𝑋: 𝑋𝐵 = 3: 1. 𝑀 is the midpoint of 𝐵𝐶.
Show that 𝑋𝑀 is parallel to 𝑂𝐶.
𝑂𝐶 = 𝑎 + 𝑏
𝑋𝑀 =
1
4
−𝑎 + 𝑏 +
1
2
𝑎 =
1
4
𝑎 +
1
4
𝑏
=
1
4
𝑎 + 𝑏
𝑋𝑀 is a multiple of 𝑂𝐶 ∴ parallel.
The key is to factor out a scalar such
that we see the same vector.
For any proof question always find the vectors
involved first, in this case 𝑋𝑀 and 𝑂𝐶.
The magic words here are
“is a multiple of”.
?
22. Test Your Understanding
𝑃
𝐵
𝑂
𝐴
2𝒂
3𝒃
a) Find 𝐴𝐵 in terms of 𝒂 and 𝒃.
−2𝒂 + 3𝒃
b) 𝑃 is the point on 𝐴𝐵 such that
𝐴𝑃: 𝑃𝐵 = 2: 3.
Show that 𝑂𝑃 is parallel to the vector
𝒂 + 𝒃.
?
Edexcel June 2011 Q26
Bro Exam Note: Notice that the mark
scheme didn’t specifically require “is a
multiple of” here (but write it anyway!), but
DID explicitly factorise out the
6
5
?
23. Proving three points form a straight line
Points A, B and C form a straight line if:
𝑨𝑩 and 𝑩𝑪 are parallel (and B is a common point).
Alternatively, we could show 𝑨𝑩 and 𝑨𝑪 are parallel. This tends to be easier.
A
B
C
?
A
B
C
24. 𝐴𝑁 = 2𝒃, 𝑁𝑃 = 𝒃
𝐵 is the midpoint of 𝐴𝐶. 𝑀 is the midpoint of 𝑃𝐵.
a) Find 𝑃𝐵 in terms of 𝒂 and 𝒃.
b) Show that 𝑁𝑀𝐶 is a straight line.
Straight Line Example
−3𝒃 + 𝒂
𝑁𝑀 = 𝑏 +
1
2
𝑎 − 3𝑏
= 𝑏 +
1
2
𝑎 −
3
2
𝑏
=
1
2
𝑎 −
1
2
𝑏
=
𝟏
𝟐
(𝒂 − 𝒃)
𝑁𝐶 = −2𝑏 + 2𝑎
= 𝟐 𝒂 − 𝒃
𝑵𝑪 is a multiple of 𝑵𝑴 ∴ 𝑵𝑪 is
parallel to 𝑵𝑴.
𝑵 is a common point.
∴ 𝑵𝑴𝑪 is a straight line.
b
a
?
?
25. November 2013 1H Q24
𝑂𝐴 = 𝑎 and 𝑂𝐵 = 𝑏
𝐷 is the point such that 𝐴𝐶 = 𝐶𝐷
The point 𝑁 divides 𝐴𝐵 in the ratio 2: 1.
(a) Write an expression for 𝑂𝑁 in terms of 𝒂 and 𝒃.
𝑶𝑵 = 𝒃 +
𝟏
𝟑
−𝒃 + 𝒂 =
𝟏
𝟑
𝒂 +
𝟐
𝟑
𝒃
(b) Prove that 𝑂𝑁𝐷 is a straight line.
𝑶𝑫 = 𝒂 + 𝟐𝒃
𝑶𝑵 =
𝟏
𝟑
(𝒂 + 𝟐𝒃)
𝑶𝑵 is a multiple of 𝑶𝑫 and 𝑶 is a common point.
∴ 𝑶𝑵𝑫 is a straight line.
?
?
Test Your Understanding
26. Exercise 3
𝑇 is the point on 𝐴𝐵 such that
𝐴𝑇: 𝑇𝐵 = 5: 1. Show that 𝑂𝑇 is
parallel to the vector 𝒂 + 2𝒃.
𝑂𝑇 = 2𝒃 +
1
6
−2𝒃 + 5𝒂
= 2𝒃 −
1
3
𝒃 +
5
6
𝒂
=
5
6
𝒂 +
5
3
𝒃
=
5
6
𝒂 + 2𝒃
1
𝑀 is the midpoint of 𝑂𝐴. 𝑁 is the
midpoint of 𝑂𝐵. Prove that 𝐴𝐵 is
parallel to 𝑀𝑁.
𝐴𝐵 = −2𝒎 + 2𝒏
= 2 −𝒎 + 𝒏
𝑀𝑁 = −𝒎 + 𝒏
𝐴𝐵 is a multiple of 𝑀𝑁 ∴ parallel.
2
?
?
27. Exercise 3
3
𝐴𝐶𝐸𝐹 is a parallelogram. 𝐵 is the
midpoint of 𝐴𝐶. 𝑀 is the midpoint of 𝐵𝐸.
Show that 𝐴𝑀𝐷 is a straight line.
𝐷𝑀 = −𝒃 +
1
2
3𝒃 + 𝒂
= −𝒃 +
3
2
𝒃 +
1
2
𝒂
=
1
2
𝒂 +
1
2
𝒃 =
1
2
𝒂 + 𝒃
𝐷𝐴 = 2𝒃 + 2𝒂 = 2 𝒂 + 𝒃
𝐷𝐴 is a multiple of 𝐷𝑀 and 𝐷 is a
common point, so 𝐴𝑀𝐷 is a straight line.
4
𝐶𝐷 = 𝒂, 𝐷𝐸 = 𝒃 and 𝐹𝐶 = 𝒂 − 𝒃
i) Express 𝐶𝐸 in terms of 𝒂 and 𝒃.
𝒂 + 𝒃
ii) Prove that 𝐹𝐸 is parallel to 𝐶𝐷.
𝐹𝐸 = 𝑎 − 𝑏 + 𝑎 + 𝑏 = 2𝒂 which
is a multiple of 𝐶𝐷
iii) 𝑋 is the point on 𝐹𝑀 such that such that
𝐹𝑋: 𝑋𝑀 = 4: 1. Prove that 𝐶, 𝑋 and 𝐸 lie on the
same straight line.
𝐶𝑋 = −𝒂 + 𝒃 +
4
5
2𝒂 −
1
2
𝒃 =
3
5
𝒂 +
3
5
𝒃
=
3
5
𝒂 + 𝒃
𝐶𝐸 = 𝒂 + 𝒃
?
?
?
?
28. Exercise 3
5
𝑂𝐴𝐵𝐶 is a parallelogram. 𝑃 is the point on
𝐴𝐶 such that 𝐴𝑃 =
2
3
𝐴𝐶.
i) Find the vector 𝑂𝑃. Give your answer in
terms of 𝒂 and 𝒄.
𝑂𝑃 = 6𝒂 +
2
3
−6𝒂 + 6𝒄
= 2𝒂 + 4𝒄 = 2 𝒂 + 2𝒄
ii) Given that the midpoint of 𝐶𝐵 is 𝑀,
prove that 𝑂𝑃𝑀 is a straight line.
𝑂𝑀 = 3𝒂 + 6𝒄 = 3 𝒂 + 2𝒄
𝑂𝑀 is a multiple of 𝑂𝑃 and 𝑂 is a common
point, therefore 𝑂𝑃𝑀 is a straight line.
6
𝑂𝐴 = 3𝒂 and 𝐴𝑄 = 𝒂 and 𝑂𝐵 = 𝒃 and
𝐵𝐶 =
1
2
𝒃. 𝑀is the midpoint of 𝑄𝐵. Prove
that 𝐴𝑀𝐶 is a straight line.
𝐴𝑀 = 𝒂 +
1
2
−4𝒂 + 𝒃 = −𝒂 +
1
2
𝒃
𝐴𝐶 = −3𝒂 +
3
2
𝒃 = 3 −𝒂 +
1
2
𝒃
𝐴𝐶 is a multiple of 𝐴𝑀 and 𝐴 is a
common point, therefore 𝐴𝑀𝐶 is a
straight line.
?
?
?