CHAPTER 29 VECTORS Learning objectives: In this chapter you will learn how to • describe a translation by using a vector, • add and subtract vectors, • multiply a vector by a scalar. Success criteria: At the end of the chapter you should be able to • understand vectors and calculate them. 29.1 Introduction to vectors Vector is a quantity that has both magnitude and direction. It is most commonly represented by a directed line segment whose length represents the magnitude and whose orientation with an arrow represents the direction. Vectors are frequently used in physics to represent velocity, acceleration, force and momentum. We denote vectors using: • bold lower cases x, • using a line underneath the lower case x (or a line over it), #» or • using an arrow above lower cases x # » • using an arrow above two upper cases AB, which represent the start point (A) and end point (B) of a vector. 1 Vectors drawn in a coordinate grid (below) can be represented by two numbers in bracketsina column. 4 # » AB = means to get from the starting point A to the end point B you need to 2 move 4 to the right and 2 up. Vectors a, b and c in the picture below have the same magnitude. Even though all 3 vectors are parallel, they all do not have the same direction. Vectors a and c have the same (arrow) direction, so the vectors are the same. Vector b has the opposite direction of vector a (and c), so it is the opposite vector. Opposite vector to vector a is denoted as -a and the opposite vector to vector # » # » # » # » AB is denoted as −AB. Therefore BA = −AB. 2 Adding and subtracting vectors Graphically: Graphically subtracting vectors is the same as adding the opposite vector: a − b = a + (-b). Algebraically adding and subtracting vectors: 4 1 4+1 5 + = = 2 −3 2 + (−3) −1 4 1 4−1 3 − = = 2 −3 2 − (−3) 5 a c a±c ± = b d b±d Multiplying vectors with a scalar A scalar is just another name for a (real) number. Multiplying a vector with a scalar means enlarging the magnitude and, if the scalar is a negative number, redirecting the vector into the opposite direction as well. Graphically: Algebraically: 4 3×4 12 3× = = 2 3×2 6 −1 −2 × (−1) 2 −2 × = = 2 −1 × 2 −4 x kx = k× y ky Practise and Homework: Chapter 29.1: all (core) exercises Set task on MyiMaths *Challenge work*: read about using vectors and their magnitude in Chapters 29.2, 29.3 and complete the exercises from the textbook. 3