Vectors in the Plane 8.3 Part 1

2  Write vectors as linear combinations of unit vectors.  Find the direction angles of vectors.  Use vectors to model and solve real-life problems. Objectives

3 Introduction

4 Scalars and Vectors A scalar is a single number that represents a magnitude –E.g. distance, mass, speed, temperature, etc. A vector is a set of numbers that describe both a magnitude and direction –E.g. velocity (the magnitude of velocity is speed), force, momentum, etc. Can be represented by an arrow v

5 Characteristics of Vectors A Vector is something that has two and only two defining characteristics: 1.Magnitude: the 'size' or 'quantity' 2.Direction: the vector is directed from one place to another.

6 Directed Line Segments Quantities such as force and velocity involve both magnitude and direction and cannot be completely characterized by a single real number. To represent such a quantity, you can use a directed line segment (vector), as shown in Figure Figure 3.11

7 Directed Line Segments The directed line segment has initial point P and terminal point Q. Its magnitude (or length) is denoted by and can be found using the Distance Formula. Two directed line segments that have the same magnitude and direction are equivalent. For example, the directed line segments in Figure 3.12 are all equivalent. Figure 3.12

8 Directed Line Segments The set of all directed line segments that are equivalent to the directed line segment is a vector v in the plane, written Vectors are denoted by lowercase, boldface letters such as u, v, and w.

9 Notation Using a single lowercase, boldface letter such as u, v, or w. Using the initial and terminal points at either end, arrow on top. A B The vector AB The vector u u

10 Component Form of a Vector

11 Component Form of a Vector The directed line segment whose initial point is the origin is often the most convenient representative of a set of equivalent directed line segments. This representative of the vector v is in standard position. A vector whose initial point is the origin (0, 0) can be uniquely represented by the coordinates of its terminal point (v 1, v 2 ). This is the component form of a vector v, written as v =  v 1, v 2 .

12 Component Form of a Vector The coordinates v 1 and v 2 are the components of v (v 1 is x component or distance in the x direction and v 2 is the y component or distance in the y direction). If both the initial point and the terminal point lie at the origin, then v is the zero vector and is denoted by 0 =  0, 0 . v =  v 1, v 2 

13 Component Form of a Vector x y A B 3 2 This vector can be written as:

14 Calculating Vectors From Points Point-point subtraction yields a vector Head point – Tail point v = P - Q

15 point - point = vector A B B – A A B A – B Head point – Tail point

16 Example Given the points; A(2, -1), B(5, 0), C(1, 2). Solution:

17 Component Form of a Vector Two vectors u =  u 1, u 2  and v =  v 1, v 2  are equal if and only if u 1 = v 1 and u 2 = v 2. For instance, the vector u from P(0, 0) to Q(3, 2) is u = =  3 – 0, 2 – 0  =  3, 2 , and the vector v from R(1, 2) to S(4, 4) is v = =  4 – 1, 4 – 2  =  3, 2 .

18 Types of Vectors Position Vector Displacement Vector

19 Position Vectors Used to define the position of a point. Always starts at the origin. The components of the column vector will always be the same as the coordinates of the point.

20 Position Vectors The vector of a point from the origin is called it’s Position Vector Every point has a unique position vector O R OR y x

21 Position Vectors A displacement vector has magnitude and direction, but is not fixed to the origin s A position vector is fixed to the origin O R OR y x

22 Displacement Vectors Differ from position vectors in that they have no specific position – just represent a change in position. A displacement vector is the difference between an object’s initial and final positions.

23 Displacement Vectors A Vector has BOTH a Length & a Direction k can be in any position k k kk All 4 Vectors here are EQUAL in Length and Travel in SAME Direction. All called k

24 Describe the vectors in component form b a c d

25 The zero vector, denoted by 0, has length 0. It is the only vector with no specific direction. ZERO VECTOR

26 Magnitude of a Vector The length of a vector u is the magnitude or modulus of u and is denoted by ‖ u ‖. The magnitude can be found using the Pythagoras’ Theorem (magnitude vector = square root of the sum of the squares of the components of the vector). The distance between two points is the magnitude of the vector. For a constant k, the length of the vector ku = ‖ ku ‖ = |k| ‖ u ‖.

27 Magnitude of Vector

28 Your Turn: Find the component form and magnitude of the vector v that has initial point (4, –7) and terminal point (–1, 5). Solution: Let P(4, –7) = (p 1, p 2 ) and Q(–1, 5) = (q 1, q 2 ). Then, the components of v =  v 1, v 2  are v 1 = q 1 – p 1 v 2 = q 2 – p 2 = –1 – 4 = –5 = 5 – (–7) = 12.

29 Solution So, v =  –5, 12  and the magnitude of v is cont’d

30 Vector Operations

31 Vector Operations The two basic vector operations are scalar multiplication and vector addition. In operations with vectors, numbers are usually referred to as scalars. In this section, scalars will always be real numbers. Geometrically, the product of a vector v and a scalar k is the vector that is | k | times as long as v.

32 Vector Operations When k is positive, kv has the same direction as v, and when k is negative, kv has the direction opposite that of v, as shown in Figure To add two vectors u and v geometrically, first position them (without changing their lengths or directions) so that the initial point of the second vector v coincides with the terminal point of the first vector u. Figure 3.14

33 Vector Operations The sum u + v is the vector formed by joining the initial point of the first vector u with the terminal point of the second vector v, as shown below.

34 Vector Operations This technique is called the parallelogram law for vector addition because the vector u + v, often called the resultant of vector addition, is the diagonal of a parallelogram having adjacent sides u and v.

35 Vector Operations The negative of v =  v 1, v 2  is –v = (–1)v =  –v 1, –v 2  and the difference of u and v is u – v = u + (–v) =  u 1 – v 1, u 2 – v 2 . Negative u – v = u + (–v) Figure 3.15 Difference Add (–v). See Figure 3.15

36 Vector Operations To represent u – v geometrically, you can use directed line segments with the same initial point. The difference u – v is the vector from the terminal point of v to the terminal point of u, which is equal to u + (–v), as shown in Figure The component definitions of vector addition and scalar multiplication are illustrated in Example 3. In this example, notice that each of the vector operations can be interpreted geometrically.

37 Example 3 – Vector Operations Let v =  –2, 5  and w =  3, 4 . Find each of the following vectors. a. 2v b. w – v c. v + 2w Solution: a. Because v =  –2, 5 , you have 2v = 2  –2, 5  =  2(–2), 2(5)  =  –4, 10 . A sketch of 2v is shown in Figure Figure 3.16

38 Example 3 – Solution b. The difference of w and v is w – v =  3, 4  –  –2, 5  =  3 – (–2), 4 – 5  =  5, –1  A sketch of w – v is shown in Figure Note that the figure shows the vector difference w – v as the sum w + (–v). Figure 3.17 cont’d

39 Example 3 – Solution c. The sum of v and 2w is v + 2w =  –2, 5  + 2  3, 4  =  –2, 5  +  2(3), 2(4)  =  –2, 5  +  6, 8  =  –2 + 6,  =  4, 13 . A sketch of v + 2w is shown in Figure Figure 3.18 cont’d

40 Vector Operations Vector addition and scalar multiplication share many of the properties of ordinary arithmetic.

41 Trigonometry Pages , 6, 10, 14, 16, 18, 20, 24, 26, 28 Chapter 3-3 Homework