MAC 1114 Module 9 Introduction to Vectors
Learning Objectives Upon completing this module, you should be able to: 1. Learn and apply basic concepts about vectors. 2. Perform operations on vectors. 3. Represent a vector quantity algebraically and find unit vectors. 4. Compute dot products and the angle between two vectors. 5. Use vectors to solve applications. 2
Introduction to Vectors There are two major topics in this module: - Introduction to Vectors, Operations, and the Dot Products - Application of Vectors 3
Quick Review on Parallel Lines and Transversal Parallel lines are lines that lie in the same plane and do not intersect. When a line q intersects two parallel lines, q, is called a transversal. Transversal q m n parallel lines 4
Important Angle Relationships q m n Name Alternate interior angles Alternate exterior angles Interior angles on the same side of the transversal Corresponding angles Angles 4 and 5 3 and 6 1 and 8 2 and 7 4 and 6 3 and 5 2 & 6, 1 & 5, 3 & 7, 4 & 8 Rule Angles measures are equal. Angle measures are equal. Angle measures add to 180. Angle measures are equal. 5
Basic Terminology A vector in the plane is a directed line segment. Consider vector AB A is called the initial point B is called the terminal point Magnitude: length of a vector, expressed as The sum of two vectors is also a vector. The vector sum A + B is called the resultant. 6
Basic Terminology Continued A vector with its initial point at the origin is called a position vector. A position vector u with its endpoint at the point (a, b) is written The numbers a and b are the horizontal component and vertical component of vector u. The positive angle between the x-axis and a position vector is the direction angle for the vector. 7
What are Magnitude and Direction Angle of Vector? The magnitude (length) of vector u = is given by The direction angle θ satisfies where a 0. 8
Example of Finding Magnitude and Direction Angle Find the magnitude and direction angle for Magnitude: Vector u has a positive horizontal component. Vector u has a negative vertical component, placing the vector in quadrant IV. Direction Angle: 9
What are the Horizontal and Vertical Components? The horizontal and vertical components, respectively, of a vector u having magnitude u and direction angle θ are given by That is, 10
Example of Finding the Horizontal and Vertical Components Vector w has magnitude 35.0 and direction angle 51.2. Find the horizontal and vertical components. Therefore, w = The horizontal component is 21.9, and the vertical component is 27.3. 11
Example Write each vector in the Figure on the right in the form u = 5cos60 o,5sin60 o 12
Solutions 13
What are Vector Operations? For any real numbers a, b, c, d, and k, 14
Example: Vector Operations Let u = 4,10 and v = 5,!3 find: a) 4v c) 2u 3v b) 2u + v 15
How to Compute Dot Product? A unit vector is a vector that has magnitude 1. Dot Product The dot product of two vectors is denoted u v, read u dot v, and given by 16
Example of Finding Dot Products Find each dot product. 17
What are the Properties of the Dot Product? For all vectors u, v, and w and real numbers k, a) b) c) d) e) f) 18
What is the Geometric Interpretation of Dot Product? If θ is the angle between the two nonzero vectors u and v, where 0 θ 180, then 19
Example of Finding the Angle Between the Two Vectors Find the angle θ between two vectors By the geometric cos! = u" v u v = 5,6 " 3,4 5,6 3,4 5(3) + 6(4) = 25+ 36 9 +16 = 39 5 61 #.9986876635! # cos $1.9986876635 # 2.94 o 20
Example Forces of 10 newtons and 50 newtons act on an object at right angles to each other. Find the magnitude of the resultant and the angle of the resultant makes with the larger force. 10 v 10 θ 50 The resultant vector, v, has magnitude 51 and make an angle of 11.3 with the larger force. 21
Example A vector w has a magnitude of 45 and rests on an incline of 20. Resolve the vector into its horizontal and vertical components. 45 v 20 u The horizontal component is 42.3 and the vertical component is 15.4. 22
Example A ship leaves port on a bearing of 28.0 and travels 8.20 mi. The ship then turns due east and travels 4.30 mi. How far is the ship from port? What is its bearing from port? 23
Example Continued Vectors PA and AE represent the ship s path. Magnitude and bearing: 24
Example Continued The ship is about 10.9 mi from port. To find the bearing of the ship from port, find angle APE. 10.9 Add 20.4 to 28.0 to find that the bearing is 48.4. 25
What is the Equilibrant? We have learned how to find the resultant of two vectors. A vector that will counterbalance the resultant is called the equilibrant. For instance, the equilibrant of vector u is the vector -u. 26
What have we learned? We have learned to: 1. Learn and apply basic concepts about vectors. 2. Perform operations on vectors. 3. Represent a vector quantity algebraically and find unit vectors. 4. Compute dot products and the angle between two vectors. 5. Use vectors to solve applications. 27
Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8th Edition 28