PHY2048 Lecture 1: Introduction to Vectors and Operations with Vectors PDF

PHY2048 Lecture 1 Introduction to Vectors and Operations with Vectors Copyright © 2020 Pearson Education, Inc. All Rights Reserved Standards and Units Length, time, and mass are three fundamental quantities of physics. The International System (SI for Système Internat...

PHY2048 Lecture 1 Introduction to Vectors and Operations with Vectors Copyright © 2020 Pearson Education, Inc. All Rights Reserved Standards and Units Length, time, and mass are three fundamental quantities of physics. The International System (SI for Système International) is the most widely used system of units. In SI units, length is measured in meters, time in seconds, and mass in kilograms. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Unit Prefixes Prefixes can be used to create larger and smaller units for the fundamental quantities. Some examples are: 1 m = 10 −6 m (size of some bacteria and living cells) 1 km = 103 m (a 10-minute walk) −6 1 mg = 10 kg (mass of a grain of salt) −3 1 g = 10 kg (mass of a paper clip) −9 1 ns = 10 s (time for light to travel 0.3 m) Copyright © 2020 Pearson Education, Inc. All Rights Reserved Unit Consistency and Conversions An equation must be dimensionally consistent. Terms to be added or equated must always have the same units. (Be sure you’re adding “apples to apples.”) Always carry units through calculations. Convert to standard units as necessary, by forming a ratio of the same physical quantity in two different units, and using it as a multiplier. For example, to find the number of seconds in 3 min, we write:  60 s  3 min = (3 min )   = 180 s  1 min  Copyright © 2020 Pearson Education, Inc. All Rights Reserved Uncertainty and Significant Figures The uncertainty of a measured quantity is indicated by its number of significant figures. For multiplication and division, the answer can have no more significant figures than the smallest number of significant figures in the factors. For addition and subtraction, the number of significant figures is determined by the term having the fewest digits to the right of the decimal point. As this train mishap illustrates, even a small percent error can have spectacular results! Copyright © 2020 Pearson Education, Inc. All Rights Reserved Vectors and Scalars A scalar quantity can be described by a single number. A vector quantity has both a magnitude and a direction in space. In this book, a vector quantity is represented in boldface italic type with an arrow over it: A. The magnitude of A is written as A or A. Copyright © 2020 Pearson Education, Inc. All Rights Reserved SCALAR Scalar Example Magnitude A SCALAR quantity Speed 35 m/s is any quantity in physics that has Distance 25 meters MAGNITUDE ONLY Age 16 years Copyright © 2020 Pearson Education, Inc. All Rights Reserved  VECTOR (Geometry) Characteristics of Vectors A Vector is a directed (with an arrow) line segment that has two and only two defining characteristics: Magnitude: the 'size' or the length of the direction is this angle segment measured from certain reference Direction: the vector is directed from one place to another. (Basically, where the arrow is pointing to, but is measured by the angle the vector makes with a predefined reference) Copyright © 2020 Pearson Education, Inc. All Rights Reserved Example The direction of the vector is 55° North of East The magnitude of the vector is 2.3. Copyright © 2020 Pearson Education, Inc. All Rights Reserved  A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector. Blue and orange Blue and purple vectors Blue and green vectors vectors have same have same magnitude have same direction but magnitude but different and direction so they different magnitude. direction. are equal. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Two vectors are equal if they have the same direction and magnitude (length). All these vectors are equal !!! Copyright © 2020 Pearson Education, Inc. All Rights Reserved VECTOR Quantity Vector quantity Magnitude and A VECTOR Example Direction quantity Velocity 35 m/s, North is any quantity in physics that has Acceleration 10 m/s2, South BOTH MAGNITUDE Displacement 20 m, East and DIRECTION Copyright © 2020 Pearson Education, Inc. All Rights Reserved Adding Two Vectors Graphically (1 of 3) Copyright © 2020 Pearson Education, Inc. All Rights Reserved Adding Two Vectors Graphically (2 of 3) Copyright © 2020 Pearson Education, Inc. All Rights Reserved Adding Two Vectors Graphically (3 of 3) Copyright © 2020 Pearson Education, Inc. All Rights Reserved Adding More Than Two Vectors Graphically (1 of 3) To add several vectors, use the head-to-tail method. The vectors can be added in any order. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Adding More Than Two Vectors Graphically (2 of 3) To add several vectors, use the head-to-tail method. The vectors can be added in any order. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Adding More Than Two Vectors Graphically (3 of 3) To add several vectors, use the head-to-tail method. The vectors can be added in any order. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Subtracting Vectors Copyright © 2020 Pearson Education, Inc. All Rights Reserved  Subtraction of Vectors. Example Determine the vector A - B by using the Tail-to-tip method A B Copyright © 2020 Pearson Education, Inc. All Rights Reserved  Subtraction of Vectors. Example Determine the vector A - B by using the parallelogram method A B Copyright © 2020 Pearson Education, Inc. All Rights Reserved Multiplying a Vector by a Scalar If c is a scalar, the product c A has magnitude c A. The figure illustrates multiplication of a vector by a) a positive scalar and b) a negative scalar. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Addition of Two Vectors at Right Angles or Copyright © 2020 Pearson Education, Inc. All Rights Reserved Representation of vectors Magnitude – Direction Components Unit Vectors Copyright © 2020 Pearson Education, Inc. All Rights Reserved Magnitude-Direction representation of vectors Copyright © 2020 Pearson Education, Inc. All Rights Reserved Magnitude-Direction representation of a vector Basically, you just give the magnitude(length) of the vector and the angle (from the positive x-axis) that the vector is pointing. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Example y A 11.7 cm ∟45° 45 ° x Copyright © 2020 Pearson Education, Inc. All Rights Reserved Example y B 11.7 cm ∟135° 135 ° x Copyright © 2020 Pearson Education, Inc. All Rights Reserved Vector Components Adding vectors graphically provides limited accuracy. Vector components provide a general method for adding vectors. Any vector can be represented by an x-component Ax and a y-component Ay. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Positive and Negative Components The components of a vector may be positive or negative numbers, as shown in the figures. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Example y A 4 A < 3, 4 > 3 x Copyright © 2020 Pearson Education, Inc. All Rights Reserved Example y B 4 B < - 3, 4 > -3 x Copyright © 2020 Pearson Education, Inc. All Rights Reserved Finding Components We can calculate the components of a vector from its magnitude and direction. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Calculations Using Components We can use the components of a vector to find its A magnitude and direction: A = Ax 2 + Ay 2 and tan = y Ax We can use the components of a set of vectors to find the components of their sum: Rx = Ax + Bx + Cx +..., Ry = Ay + By + Cy +... Copyright © 2020 Pearson Education, Inc. All Rights Reserved Unit Vectors A unit vector has a magnitude of 1 with no units. The unit vector iˆ points in the +x-direction, ĵ points in the +y-direction, and k̂ points in the +z-direction. Any vector can be expressed in terms of its components as A = Ax iˆ + Ay jˆ + Az kˆ. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Addition of vectors (Analytical Approach) Copyright © 2020 Pearson Education, Inc. All Rights Reserved Addition of vectors (Analytical Approach) Copyright © 2020 Pearson Education, Inc. All Rights Reserved The Scalar Product (1 of 2) Copyright © 2020 Pearson Education, Inc. All Rights Reserved The Scalar Product (2 of 2) The scalar product can be positive, negative, or zero, depending on the angle between A and B. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Calculating a Scalar Product Using Components In terms of components: The scalar product of two vectors is the sum of the products of their respective components. Copyright © 2020 Pearson Education, Inc. All Rights Reserved Calculating a Scalar Product Using Components Find the Scalar Product of the following vectors Copyright © 2020 Pearson Education, Inc. All Rights Reserved Finding an Angle Using the Scalar Product Copyright © 2020 Pearson Education, Inc. All Rights Reserved Scalar Product in Physics Work done by a Force Mechanical Power Copyright © 2020 Pearson Education, Inc. All Rights Reserved The Vector Product If the vector product (“cross product”) of two vectors is The direction of the vector product can be found using the right-hand rule: Copyright © 2020 Pearson Education, Inc. All Rights Reserved The Vector Product is Anticommutative A  B = −B  A Copyright © 2020 Pearson Education, Inc. All Rights Reserved Calculating the Vector Product Copyright © 2020 Pearson Education, Inc. All Rights Reserved Vector Cross Product Some properties of the cross product: Copyright © 2020 Pearson Education, Inc. All Rights Reserved