C4: Chapter 5 – Vectors Dr J Frost Last modified: 14 th April 2016

RECAP: GCSE stuff A vector has both direction and magnitude. It represents a movement. VectorCoordinate ? ? Vectors that are equal have both the same magnitude and direction. ? ?

Basics #1: Magnitude of a Vector 3 4 ? ?? ? ?

Basics #2: Scalars  A scalar is a quantity which scales a vector (hence the name). It is just an ordinary number. scalar Scalars are also used in the context of directionless quantities, e.g. velocity is a vector while speed is a scalar.

Basics #3: Unit Vectors  A unit vector is a vector whose magnitude is 1 There’s certain operations on vectors that require the vectors to be ‘unit’ vectors. We just scale the vector so that its magnitude is now 1. ?? ? ?

Basics #4: Parallel Vectors ?

Basics #5: Uniqueness of composition ? ?

?

Basics #6: Position Vectors ? It effectively allows us to treat a point a vector. ? ? ?

Basics #7: Writing Cartesian components ? ? ? Bro Tip: I really hate this notation, and just immediately write any vector in the usual form before solving more complex questions. ?

Test Your Understanding So Far 1 2 a b c ? ? ? ?

Straight Lines ? ? ? ?

Straight lines between two points ? ?

Examples ? ? ? ? ? ? 1 2 3

Test Your Understanding ? ?

Exercises Exercise 5H Q1, 3, 5 Exercise 5I Q1-5

Exam Topics I went through a large number of C4 exams and looked at the vectors questions and their subparts. I found the following topics: TopicFrequency Points of intersection/showing lines do or don’t meet.6 Angle between two lines4 Finding length of vectors/distance between two points4 Finding missing components of points on a line.2 Finding nearest point on a line to the origin.2 Finding nearest point on a line to a general point.2 Dealing with perpendicular lines/showing lines a perpendicular2 Finding angles between general vectors1 Find missing point in a parallelogram1 Show a point lies on a line1 Show 3 points are collinear1 Find the area of a rectangle formed by vectors1 Find the area of a triangle formed by vectors1 Find equation of a line1 Reflection of a point in a line1

Dot Product ? ? ?

Angle between two vectors Remarkably, if the two vectors are unit vectors, the dot product gives us the cosine of the angle between them. ?

Perpendicular vectors Using the equation from the previous slide… ? ?

Angles between straight lines i.e. we only care about the directional part of the line, not how we got to the line. ? ? ? ?

Test Your Understanding ? ? ? ? ? ? (Hint: a parallelogram consists of two non-right angled triangles)

Exercises Exercise 5G Q1, 3, 5, 11, 12

Exam Topics TopicFrequency Points of intersection/showing lines do or don’t meet.6 Angle between two lines4 Finding length of vectors/distance between two points4 Finding missing components of points on a line.2 Finding nearest point on a line to the origin.2 Finding nearest point on a line to a general point.2 Dealing with perpendicular lines/showing lines a perpendicular2 Finding angles between general vectors1 Find missing point in a parallelogram1 Show a point lies on a line1 Show 3 points are collinear1 Find the area of a rectangle formed by vectors1 Find the area of a triangle formed by vectors1 Find equation of a line1 Reflection of a point in a line1

Finding nearest point on a line to a point This one is not in your textbook! ? ? ? ??

Test Your Understanding C4 Jan 2013 Q7 ?

Moving a given distance in a given direction Bro Tip: If you convert a vector to a unit vector, then moving by it will clearly move a distance of 1. ?

Test Your Understanding ?