What are vectors. Adding Vectors
|
|
- Liliana Horton
- 5 years ago
- Views:
Transcription
1 Vectors Introduction What are vectors Each vector is defined by two pieces of information: Direction and Magnitude. Often vectors are described by a picture representation or by ordered pairs which describe the direction and magnitude of the vector. To distinguish from the ordered pairs describing a point, vectors are written using pointy brackets rather than parenthesis. Variables representing vectors are often written in bold or with a hat or arrow over them. Here are a few examples. Examples of Vectors < 1,2 > < 2,4 > < 1, 2 > < 3,1 > Adding Vectors To add two vectors, we simply add the corresponding components together. Using the diagrams to represent vectors we add by placing one vector at the tip of another, tip-to-tail. The vector uniting the starting point to the end point is commonly referred to as the resultant vector or the sum of the vectors. In some sense, the picture of addition of two vectors shows why vectors were invented to begin with. Their roots are in physics, perhaps the most famous example of a vector is a vector representing some Force. because Force is composed to two pieces of information, direction and magnitude, vectors are tailor made to represent such concepts. The direction of the force is represented by the direction of the vector while the magnitude of the force is represented by the magnitude of the vector. With this in mind, adding vectors works just as you may expect if you were adding two forces, tip-to-tail on the picture, component-wise algebraically. For Example: < 1, 2 > + < 3, 1 >=< 4, 3 > w =< 3,1 > v =< 1,2 > v + w =< 4, 3 > Normalizing of Vectors To normalize a vector v refers to producing a new vector n v such that this new vector has the same direction as v BUT magnitude exactly equal to one. Any non-zero vector can be normalized by simply multiplying by a scalar. Such scalar is 1/magnitude v. In other words, n v = 1 v v Subtracting Vectors The key idea to subtract vectors is to turn the subtraction question into an addition question. For example suppose we want to find v w. We can call such vector x = v w. Then we add w to both sides to get x + w = v Using the diagrams for each vector we can put v and w, tail-to tail and solve for x as follows. v =< 1,2 > x =< 2,1 > w =< 3,1 >
2 Scalar Multiples of Vectors Generally, we define the scalar of a vector to be a vector with the same direction (or opposite direction) and possible scaled to a different size. If c is a real number and v =< a, b > a vector, then cv = c < a, b >=< ca, cb > is referred to as v times the scalar c. Note, if the scalar is negative, it reverses the direction of the vector. Here are some examples of v =< 2, 1 > v =< 2, 1 > scalars time a vector v 2v =< 4,2 > 1 2 v =< 1, 1 2 > Magnitude of Vectors Size, magnitude, and norm of a vector are often used interchangeably. In symbols, we use the double bars to denote the magnitude of a vector. v = norm of v To calculate the norm of a vector it is often helpful to draw a picture and pythagoras (i just made it a verb!) away 0 : w =< 3,2 > w =
3 1. Vector Arithmetic (a) Vector ADDITION Suppose = <1, 2> and w = < 3, 5>, compute + w + w = <1, 2> + < 3, 5> = <1 + 3, 2 + 5> (definition of addition on vectors) = < 2, 7> (by inspection) + w w (b) Vector ADDITION Suppose = <1, 3> and w = <1, 6>, compute + w + w = <1, 3> + <1, 6> = <1 + 1, 3 + 6> (definition of addition on vectors) = <2, 3> (by inspection)
4 Trigonometry w + w (c) Vector ADDITION Suppose = <1, 1> and w = <2, 3>, compute + w + w = <1, 1> + <2, 3> = <1 + 2, 1 + 3> (definition of addition on vectors) = <3, 2> (by inspection) w + w (d) Vector ADDITION Suppose = <5, 2> and w = < 3, 1>, compute + w
5 + w = <5, 2> + < 3, 1> = <5 + 3, 2 + 1> (definition of addition on vectors) = <2, 3> (by inspection) + w w (e) Vector ADDITION Suppose = <5, 1> and w = <1, 2>, compute + w + w = <5, 1> + <1, 2> = <5 + 1, 1 + 2> (definition of addition on vectors) = <6, 1> (by inspection)
6 + w w (f) Vector ADDITION Suppose = <6, 2> and w = <2, 3>, compute + w + w = <6, 2> + <2, 3> = <6 + 2, 2 + 3> (definition of addition on vectors) = <8, 5> (by inspection) w + w (g) Vector ADDITION Suppose = <5, 1> and w = <1, 2>, compute + w
7 + w = <5, 1> + <1, 2> = <5 + 1, 1 + 2> (definition of addition on vectors) = <6, 1> (by inspection) + w w (h) Vector ADDITION Suppose = < 3, 2> and w = < 1, 5>, compute + w + w = < 3, 2> + < 1, 5> = < 3 + 1, 2 + 5> (definition of addition on vectors) = < 4, 7> (by inspection)
8 w + w 2. Vector Arithmetic (a) Vector SUBTRACTION Suppose = <1, 2> and w = < 3, 5>, compute w w = <1, 2> < 3, 5> = <1 3, 2 5> (definition of subtraction on vectors) = <4, 3> (by inspection) w w (b) Vector SUBTRACTION Suppose = <1, 3> and w = <1, 6>, compute w
9 Trigonometry w = <1, 3> <1, 6> = <1 1, 3 6> (definition of subtraction on vectors) = <0, 9> (by inspection) w w (c) Vector SUBTRACTION Suppose = <1, 1> and w = <2, 3>, compute w w = <1, 1> <2, 3> = <1 2, 1 3> (definition of subtraction on vectors) = < 1, 4> (by inspection)
10 w w (d) Vector SUBTRACTION Suppose = <5, 2> and w = < 3, 1>, compute w w = <5, 2> < 3, 1> = <5 3, 2 1> (definition of subtraction on vectors) = <8, 1> (by inspection) w w (e) Vector SUBTRACTION Suppose = <5, 1> and w = <1, 2>, compute w
11 w = <5, 1> <1, 2> = <5 1, 1 2> (definition of subtraction on vectors) = <4, 3> (by inspection) w w (f) Vector SUBTRACTION Suppose = <6, 2> and w = <2, 3>, compute w w = <6, 2> <2, 3> = <6 2, 2 3> (definition of subtraction on vectors) = <4, 1> (by inspection)
12 w w (g) Vector SUBTRACTION Suppose = <5, 1> and w = <1, 2>, compute w w = <5, 1> <1, 2> = <5 1, 1 2> (definition of subtraction on vectors) = <4, 3> (by inspection) w w (h) Vector SUBTRACTION Suppose = < 3, 2> and w = < 1, 5>, compute w
13 w = < 3, 2> < 1, 5> = < 3 1, 2 5> (definition of subtraction on vectors) = < 2, 3> (by inspection) w w 3. Vector Arithmetic (a) Vector SCALARS Suppose = <2, 0>compute 2 2 = 2<2, 0> = <2(2), 2(0)> (def of scalar multiplication) = <4, 0> (by inspection)
14 2 (b) Vector SCALARS Suppose = <2, 0>compute 3 3 = 3<2, 0> = <3(2), 3(0)> (def of scalar multiplication) = <6, 0> (by inspection) 3 (c) Vector SCALARS Suppose = <2, 0>compute
15 2 = 2<2, 0> = < 2(2), 2(0)> (def of scalar multiplication) = < 4, 0> (by inspection) 2 (d) Vector SCALARS Suppose = <4, 2>compute 2 2 = 2<4, 2> = <2(4), 2(2)> (def of scalar multiplication) = <8, 4> (by inspection)
16 2 (e) Vector SCALARS Suppose = <4, 2>compute 3 3 = 3<4, 2> = <3(4), 3( 2)> (def of scalar multiplication) = <12, 6> (by inspection) 3 (f) Vector SCALARS Suppose = <1, 3>compute
17 Trigonometry 2 = 2<1, 3> = <2(1), 2(3)> (def of scalar multiplication) = <2, 6> (by inspection) 2 (g) Vector SCALARS Suppose = <1, 3>compute 3 3 = 3<1, 3> = <3(1), 3(3)> (def of scalar multiplication) = <3, 9> (by inspection)
18 Trigonometry 3 (h) Vector SCALARS Suppose = <1, 3>compute = 1.5<1, 3> = <1.5(1), 1.5(3)> (def of scalar multiplication) = <1.5, 4.5> (by inspection) 1.5 (i) Vector SCALARS Suppose = <1, 3>compute
19 Trigonometry.75 =.75<1, 3> = <.75(1),.75(3)> (def of scalar multiplication) = <0.75, 2.25> (by inspection).75 (j) Vector SCALARS Suppose = <4, 2>compute 1 1 = 1<4, 2> = < 1(4), 1(2)> (def of scalar multiplication) = < 4, 2> (by inspection)
20 1 4. Vector Arithmetic: Famous Vectors There are two very famous vectors, there are: i =< 1, 0 > and j =< 0, 1 >. (a) FAMOUS VECTORS i and j Compute and draw the following vectors 4i + 2j let = 4i + 2j = 4 < 1, 0 > +2 < 0, 1 > =< 4, 0 > + < 0, 2 > (def of scalar multiplication) =< 4, 2 > 4i + 2j 2j 4i
21 (b) FAMOUS VECTORS i and j Compute and draw the following vectors 1i + 2j let = 1i + 2j = 1 < 1, 0 > + 2 < 0, 1 > =< 1, 0 > + < 0, 2 > (def of scalar multiplication) =< 1, 2 > 1i 1i + 2j 2j (c) FAMOUS VECTORS i and j Compute and draw the following vectors 5i + 2j let = 5i + 2j = 5 < 1, 0 > + 2 < 0, 1 > =< 5, 0 > + < 0, 2 > (def of scalar multiplication) =< 5, 2 >
22 5i 5i + 2j 2j (d) FAMOUS VECTORS i and j Compute and draw the following vectors 5i + 1j let = 5i + 1j = 5 < 1, 0 > +1 < 0, 1 > =< 5, 0 > + < 0, 1 > (def of scalar multiplication) =< 5, 1 > 5i + 1j 5i 1j (e) FAMOUS VECTORS i and j Compute and draw the following vectors 6i + 3j
23 let = 6i + 3j = 6 < 1, 0 > +3 < 0, 1 > =< 6, 0 > + < 0, 3 > (def of scalar multiplication) =< 6, 3 > 6i + 3j 3j 6i (f) FAMOUS VECTORS i and j Compute and draw the following vectors 3i + 4j let = 3i + 4j = 3 < 1, 0 > +4 < 0, 1 > =< 3, 0 > + < 0, 4 > (def of scalar multiplication) =< 3, 4 >
24 3i + 4j 4j 3i 5. NORM of a VECTOR Find the norm of the indicated vector. (a) Compute: < 4, 2 > < 4, 2 > = (4) 2 + (2) 2 (def of norm) = 20 (by Inspection) 4.47 (by Calc) (b) Compute:
25 < 1, 2 > < 1, 2 > = (1) 2 + ( 2) 2 (def of norm) = 5 (by Inspection) 2.24 (by Calc) (c) Compute: < 5, 2 > < 5, 2 > = (5) 2 + ( 2) 2 (def of norm) = 29 (by Inspection) 5.39 (by Calc)
26 (d) Compute: < 5, 1 > < 5, 1 > = (5) 2 + (1) 2 (def of norm) = 26 (by Inspection) 5.1 (by Calc) (e) Compute: < 6, 3 >
27 < 6, 3 > = (6) 2 + (3) 2 (def of norm) = 45 (by Inspection) 6.71 (by Calc) (f) Compute: < 5, 0 > < 5, 0 > = (5) 2 + (0) 2 (def of norm) = 25 (by Inspection) 5 (by Calc)
28 Which is Bigger? (a) Determine which number is larger: 4i + 2j OR ( 4i + 2j ) 4i + 2j = < 4, 2 > (def of i and j) = (4) 2 + (2) 2 (def of norm) = 20 (by Inspection) 4.47 (by Calc) Meanwhile 4i + 2j = < 4, 0 > + < 0, 2 > (def of i and j) = (def of norm) = 6 (by Inspection) Therefore, 4i + 2j is larger. (b) Determine which number is larger: 1i + 2j OR ( 1i + 2j ) 1i + 2j = < 1, 2 > (def of i and j) = (1) 2 + ( 2) 2 (def of norm) = 5 (by Inspection) 2.24 (by Calc) Meanwhile
29 1i + 2j = < 1, 0 > + < 0, 2 > (def of i and j) = (def of norm) = 3 (by Inspection) Therefore, 1i + 2j is larger. (c) Determine which number is larger: 5i + 2j OR ( 5i + 2j ) 5i + 2j = < 5, 2 > (def of i and j) = (5) 2 + ( 2) 2 (def of norm) = 29 (by Inspection) 5.39 (by Calc) Meanwhile 5i + 2j = < 5, 0 > + < 0, 2 > (def of i and j) = (def of norm) = 7 (by Inspection) Therefore, 5i + 2j is larger. (d) Determine which number is larger: 5i + 1j OR ( 5i + 1j ) 5i + 1j = < 5, 1 > (def of i and j) = (5) 2 + (1) 2 (def of norm) = 26 (by Inspection) 5.1 (by Calc) Meanwhile 5i + 1j = < 5, 0 > + < 0, 1 > (def of i and j) = (def of norm) = 6 (by Inspection) Therefore, 5i + 1j is larger. (e) Determine which number is larger: 6i + 3j OR ( 6i + 3j )
30 6i + 3j = < 6, 3 > (def of i and j) = (6) 2 + (3) 2 (def of norm) = 45 (by Inspection) 6.71 (by Calc) Meanwhile 6i + 3j = < 6, 0 > + < 0, 3 > (def of i and j) = (def of norm) = 9 (by Inspection) Therefore, 6i + 3j is larger. (f) Determine which number is larger: 5i + 0j OR ( 5i + 0j ) 5i + 0j = < 5, 0 > (def of i and j) = (5) 2 + (0) 2 (def of norm) = 25 (by Inspection) 5 (by Calc) Meanwhile 5i + 0j = < 5, 0 > + < 0, 0 > (def of i and j) = (def of norm) = 5 (by Inspection) Therefore, 5i + 0j is larger. (g) u + v OR ( u + v ) the idea is to study the above pattern and understand that it will always be the case that the sum of the individual norms is larger or equal to the norm of the sum of the vectors. In some sense, this is equivalent to saying that the sum of the lengths of any two sides of a [Euclidean] triangle have to be larger than the size of the length of the the third side of the triangle. This is very famous, it is called the triangle inequality. 7. Normalize this.. Find the normalized vector for each: (a) NORMALIZE them Compute and draw the corresponding normalized vector: =< 5, 2 >
31 First we find the norm of the vector: = < 5, 2 > = (5) 2 + (2) 2 (def of norm) = 29 (by Inspection) 5.39 (by Calc) Then we scale the original vector by multiplying by 1. Let us denote the normalized vector as n n = 1 <5, 2> 5 =, , , 0.37 (def of scalar multiplication) (approximate) (by inspection) Now, NOTE: and n have the same direction, BUT, n has norm 1 as intended. (also keep in mind position does not matter for vectors.. only direction and size, thus nothing should be interpreted from their position, only their size and direction) n (b) NORMALIZE them Compute and draw the corresponding normalized vector: =< 5, 2 > First we find the norm of the vector: = < 5, 2 > = (5) 2 + ( 2) 2 (def of norm) = 29 (by Inspection) 5.39 (by Calc)
32 Then we scale the original vector by multiplying by 1. Let us denote the normalized vector as n n = 1 <5, 2> 5 =, , , 0.37 (def of scalar multiplication) (approximate) (by inspection) Now, NOTE: and n have the same direction, BUT, n has norm 1 as intended. (also keep in mind position does not matter for vectors.. only direction and size, thus nothing should be interpreted from their position, only their size and direction) n (c) NORMALIZE them Compute and draw the corresponding normalized vector: =< 3, 4 > First we find the norm of the vector: = < 3, 4 > = ( 3) 2 + (4) 2 (def of norm) = 25 (by Inspection) 5 (by Calc) Then we scale the original vector by multiplying by 1. Let us denote the normalized vector as n
33 n Trigonometry n = 1 < 3, 4> 3 =, 4 3 5, , 0.8 (def of scalar multiplication) (approximate) (by inspection) Now, NOTE: and n have the same direction, BUT, n has norm 1 as intended. (also keep in mind position does not matter for vectors.. only direction and size, thus nothing should be interpreted from their position, only their size and direction) (d) NORMALIZE them Compute and draw the corresponding normalized vector: =< 3, 4 > First we find the norm of the vector: = < 3, 4 > = ( 3) 2 + ( 4) 2 (def of norm) = 25 (by Inspection) 5 (by Calc) Then we scale the original vector by multiplying by 1. Let us denote the normalized vector as n
34 n = 1 < 3, 4> 3 =, 4 3 5, , 0.8 (def of scalar multiplication) (approximate) (by inspection) Now, NOTE: and n have the same direction, BUT, n has norm 1 as intended. (also keep in mind position does not matter for vectors.. only direction and size, thus nothing should be interpreted from their position, only their size and direction) n (e) NORMALIZE them Compute and draw the corresponding normalized vector: =< 3, 1 > First we find the norm of the vector: = < 3, 1 > = ( 3) 2 + ( 1) 2 (def of norm) = 10 (by Inspection) 3.16 (by Calc) Then we scale the original vector by multiplying by 1. Let us denote the normalized vector as n
35 n = 1 < 3, 1> 3 =, , , 0.32 (def of scalar multiplication) (approximate) (by inspection) Now, NOTE: and n have the same direction, BUT, n has norm 1 as intended. (also keep in mind position does not matter for vectors.. only direction and size, thus nothing should be interpreted from their position, only their size and direction) n (f) =< a, b > (not both zero..) n = 1 <a, b> a =, b a = a2 + b, b 2 a2 + b 2 (def of scalar multiplication) (def of norm)
Day 1: Introduction to Vectors + Vector Arithmetic
Day 1: Introduction to Vectors + Vector Arithmetic A is a quantity that has magnitude but no direction. You can have signed scalar quantities as well. A is a quantity that has both magnitude and direction.
More informationChapter 2 A Mathematical Toolbox
Chapter 2 Mathematical Toolbox Vectors and Scalars 1) Scalars have only a magnitude (numerical value) Denoted by a symbol, a 2) Vectors have a magnitude and direction Denoted by a bold symbol (), or symbol
More informationVECTORS. Given two vectors! and! we can express the law of vector addition geometrically. + = Fig. 1 Geometrical definition of vector addition
VECTORS Vectors in 2- D and 3- D in Euclidean space or flatland are easy compared to vectors in non- Euclidean space. In Cartesian coordinates we write a component of a vector as where the index i stands
More informationGeneral Physics I, Spring Vectors
General Physics I, Spring 2011 Vectors 1 Vectors: Introduction A vector quantity in physics is one that has a magnitude (absolute value) and a direction. We have seen three already: displacement, velocity,
More informationReview: Linear and Vector Algebra
Review: Linear and Vector Algebra Points in Euclidean Space Location in space Tuple of n coordinates x, y, z, etc Cannot be added or multiplied together Vectors: Arrows in Space Vectors are point changes
More informationDistance in the Plane
Distance in the Plane The absolute value function is defined as { x if x 0; and x = x if x < 0. If the number a is positive or zero, then a = a. If a is negative, then a is the number you d get by erasing
More information11.1 Vectors in the plane
11.1 Vectors in the plane What is a vector? It is an object having direction and length. Geometric way to represent vectors It is represented by an arrow. The direction of the arrow is the direction of
More informationVectors. (same vector)
Vectors Our very first topic is unusual in that we will start with a brief written presentation. More typically we will begin each topic with a videotaped lecture by Professor Auroux and follow that with
More informationVectors. Vectors. Vectors. Reminder: Scalars and Vectors. Vector Practice Problems: Odd-numbered problems from
Vectors Vector Practice Problems: Odd-numbered problems from 3.1-3.21 Reminder: Scalars and Vectors Vector: Scalar: A number (magnitude) with a direction. Just a number. I have continually asked you, which
More informationVector components and motion
Vector components and motion Objectives Distinguish between vectors and scalars and give examples of each. Use vector diagrams to interpret the relationships among vector quantities such as force and acceleration.
More information10.2 Introduction to Vectors
Arkansas Tech University MATH 2934: Calculus III Dr. Marcel B Finan 10.2 Introduction to Vectors In the previous calculus classes we have seen that the study of motion involved the introduction of a variety
More information3.1 Inequalities - Graphing and Solving
3.1 Inequalities - Graphing and Solving When we have an equation such as x = 4 we have a specific value for our variable. With inequalities we will give a range of values for our variable. To do this we
More informationLinear Algebra. 1.1 Introduction to vectors 1.2 Lengths and dot products. January 28th, 2013 Math 301. Monday, January 28, 13
Linear Algebra 1.1 Introduction to vectors 1.2 Lengths and dot products January 28th, 2013 Math 301 Notation for linear systems 12w +4x + 23y +9z =0 2u + v +5w 2x +2y +8z =1 5u + v 6w +2x +4y z =6 8u 4v
More informationSECTION 6.3: VECTORS IN THE PLANE
(Section 6.3: Vectors in the Plane) 6.18 SECTION 6.3: VECTORS IN THE PLANE Assume a, b, c, and d are real numbers. PART A: INTRO A scalar has magnitude but not direction. We think of real numbers as scalars,
More informationLesson ACTIVITY: Tree Growth
Lesson 3.1 - ACTIVITY: Tree Growth Obj.: use arrow diagrams to represent expressions. evaluate expressions. write expressions to model realworld situations. Algebraic expression - A symbol or combination
More informationNorth by Northwest - An Introduction to Vectors
HPP A9 North by Northwest - An Introduction to Vectors Exploration GE 1. Let's suppose you and a friend are standing in the parking lot near the Science Building. Your friend says, "I am going to run at
More informationVector Basics, with Exercises
Math 230 Spring 09 Vector Basics, with Exercises This sheet is designed to follow the GeoGebra Introduction to Vectors. It includes a summary of some of the properties of vectors, as well as homework exercises.
More information2- Scalars and Vectors
2- Scalars and Vectors Scalars : have magnitude only : Length, time, mass, speed and volume is example of scalar. v Vectors : have magnitude and direction. v The magnitude of is written v v Position, displacement,
More informationVectors for Beginners
Vectors for Beginners Leo Dorst September 6, 2007 1 Three ways of looking at linear algebra We will always try to look at what we do in linear algebra at three levels: geometric: drawing a picture. This
More informationVector Arithmetic and Geometry
Vector Arithmetic and Geometry In applied mathematics and physics and engineering, vectors often have two components to represent for example planar motion or more likely have three components to represent
More informationI&C 6N. Computational Linear Algebra
I&C 6N Computational Linear Algebra 1 Lecture 1: Scalars and Vectors What is a scalar? Computer representation of a scalar Scalar Equality Scalar Operations Addition and Multiplication What is a vector?
More informationVector Algebra August 2013
Vector Algebra 12.1 12.2 28 August 2013 What is a Vector? A vector (denoted or v) is a mathematical object possessing both: direction and magnitude also called length (denoted ). Vectors are often represented
More informationVECTORS vectors & scalars vector direction magnitude scalar only magnitude
VECTORS Physical quantities are classified in two big classes: vectors & scalars. A vector is a physical quantity which is completely defined once we know precisely its direction and magnitude (for example:
More informationChapter 2 One-Dimensional Kinematics
Review: Chapter 2 One-Dimensional Kinematics Description of motion in one dimension Copyright 2010 Pearson Education, Inc. Review: Motion with Constant Acceleration Free fall: constant acceleration g =
More informationVectors and their uses
Vectors and their uses Sharon Goldwater Institute for Language, Cognition and Computation School of Informatics, University of Edinburgh DRAFT Version 0.95: 3 Sep 2015. Do not redistribute without permission.
More informationVectors. For physics and calculus students. Prepared by Larry Friesen and Anne Gillis
Vectors For physics and calculus students Prepared by Larry Friesen and Anne Gillis Butler Community College http://www.butlercc.edu Vectors This project is a direct result of math/physics instructional
More informationMatrix Algebra: Vectors
A Matrix Algebra: Vectors A Appendix A: MATRIX ALGEBRA: VECTORS A 2 A MOTIVATION Matrix notation was invented primarily to express linear algebra relations in compact form Compactness enhances visualization
More informationVectors. A vector is usually denoted in bold, like vector a, or sometimes it is denoted a, or many other deviations exist in various text books.
Vectors A Vector has Two properties Magnitude and Direction. That s a weirder concept than you think. A Vector does not necessarily start at a given point, but can float about, but still be the SAME vector.
More informationFactorizing Algebraic Expressions
1 of 60 Factorizing Algebraic Expressions 2 of 60 Factorizing expressions Factorizing an expression is the opposite of expanding it. Expanding or multiplying out a(b + c) ab + ac Factorizing Often: When
More informationWe know how to identify the location of a point by means of coordinates: (x, y) for a point in R 2, or (x, y,z) for a point in R 3.
Vectors We know how to identify the location of a point by means of coordinates: (x, y) for a point in R 2, or (x, y,z) for a point in R 3. More generally, n-dimensional real Euclidean space R n is the
More informationLINEAR ALGEBRA - CHAPTER 1: VECTORS
LINEAR ALGEBRA - CHAPTER 1: VECTORS A game to introduce Linear Algebra In measurement, there are many quantities whose description entirely rely on magnitude, i.e., length, area, volume, mass and temperature.
More informationDefinition of geometric vectors
Roberto s Notes on Linear Algebra Chapter 1: Geometric vectors Section 2 of geometric vectors What you need to know already: The general aims behind the concept of a vector. What you can learn here: The
More informationChapter 7.4: Vectors
Chapter 7.4: Vectors In many mathematical applications, quantities are determined entirely by their magnitude. When calculating the perimeter of a rectangular field, determining the weight of a box, or
More informationPrecalculus. Precalculus Higher Mathematics Courses 85
Precalculus Precalculus combines the trigonometric, geometric, and algebraic techniques needed to prepare students for the study of calculus, and strengthens students conceptual understanding of problems
More information2.1 Definition. Let n be a positive integer. An n-dimensional vector is an ordered list of n real numbers.
2 VECTORS, POINTS, and LINEAR ALGEBRA. At first glance, vectors seem to be very simple. It is easy enough to draw vector arrows, and the operations (vector addition, dot product, etc.) are also easy to
More informationEuclidean Spaces. Euclidean Spaces. Chapter 10 -S&B
Chapter 10 -S&B The Real Line: every real number is represented by exactly one point on the line. The plane (i.e., consumption bundles): Pairs of numbers have a geometric representation Cartesian plane
More informationGraphical Analysis; and Vectors
Graphical Analysis; and Vectors Graphs Drawing good pictures can be the secret to solving physics problems. It's amazing how much information you can get from a diagram. We also usually need equations
More informationVectors and the Geometry of Space
Vectors and the Geometry of Space Many quantities in geometry and physics, such as area, volume, temperature, mass, and time, can be characterized by a single real number scaled to appropriate units of
More informationMathematics Standards for High School Precalculus
Mathematics Standards for High School Precalculus Precalculus is a rigorous fourth-year launch course that prepares students for college and career readiness and intended specifically for those students
More information6. Vectors. Given two points, P 0 = (x 0, y 0 ) and P 1 = (x 1, y 1 ), a vector can be drawn with its foot at P 0 and
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationCourse Notes Math 275 Boise State University. Shari Ultman
Course Notes Math 275 Boise State University Shari Ultman Fall 2017 Contents 1 Vectors 1 1.1 Introduction to 3-Space & Vectors.............. 3 1.2 Working With Vectors.................... 7 1.3 Introduction
More informationInequalities - Solve and Graph Inequalities
3.1 Inequalities - Solve and Graph Inequalities Objective: Solve, graph, and give interval notation for the solution to linear inequalities. When we have an equation such as x = 4 we have a specific value
More information10.1 Vectors. c Kun Wang. Math 150, Fall 2017
10.1 Vectors Definition. A vector is a quantity that has both magnitude and direction. A vector is often represented graphically as an arrow where the direction is the direction of the arrow, and the magnitude
More informationStatics. Today Introductions Review Course Outline and Class Schedule Course Expectations Chapter 1 ENGR 1205 ENGR 1205
Statics ENGR 1205 Kaitlin Ford kford@mtroyal.ca B175 Today Introductions Review Course Outline and Class Schedule Course Expectations Start Chapter 1 1 the goal of this course is to develop your ability
More informationIf two sides of a triangle are congruent, then it is an isosceles triangle.
1. What is the hypothesis of the conditional statement If two sides of a triangle are congruent, then it is an isosceles triangle. two sides of a triangle are congruent it is an isosceles triangle If two
More informationMathematics Revision Guide. Algebra. Grade C B
Mathematics Revision Guide Algebra Grade C B 1 y 5 x y 4 = y 9 Add powers a 3 a 4.. (1) y 10 y 7 = y 3 (y 5 ) 3 = y 15 Subtract powers Multiply powers x 4 x 9...(1) (q 3 ) 4...(1) Keep numbers without
More informationCHAPTER 2: VECTORS IN 3D
CHAPTER 2: VECTORS IN 3D 2.1 DEFINITION AND REPRESENTATION OF VECTORS A vector in three dimensions is a quantity that is determined by its magnitude and direction. Vectors are added and multiplied by numbers
More informationFunction Operations and Composition of Functions. Unit 1 Lesson 6
Function Operations and Composition of Functions Unit 1 Lesson 6 Students will be able to: Combine standard function types using arithmetic operations Compose functions Key Vocabulary: Function operation
More informationVectors. Introduction. Prof Dr Ahmet ATAÇ
Chapter 3 Vectors Vectors Vector quantities Physical quantities that have both n u m e r i c a l a n d d i r e c t i o n a l properties Mathematical operations of vectors in this chapter A d d i t i o
More informationMath 8 Notes Units 1B: One-Step Equations and Inequalities
Math 8 Notes Units 1B: One-Step Equations and Inequalities Solving Equations Syllabus Objective: (1.10) The student will use order of operations to solve equations in the real number system. Equation a
More informationVectors. Introduction
Chapter 3 Vectors Vectors Vector quantities Physical quantities that have both numerical and directional properties Mathematical operations of vectors in this chapter Addition Subtraction Introduction
More information(, ) : R n R n R. 1. It is bilinear, meaning it s linear in each argument: that is
17 Inner products Up until now, we have only examined the properties of vectors and matrices in R n. But normally, when we think of R n, we re really thinking of n-dimensional Euclidean space - that is,
More information3 Vectors. 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan
Chapter 3 Vectors 3 Vectors 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 2 3 3-2 Vectors and Scalars Physics deals with many quantities that have both size and direction. It needs a special mathematical
More informationVectors Part 1: Two Dimensions
Vectors Part 1: Two Dimensions Last modified: 20/02/2018 Links Scalars Vectors Definition Notation Polar Form Compass Directions Basic Vector Maths Multiply a Vector by a Scalar Unit Vectors Example Vectors
More informationOhio s Learning Standards-Extended. Mathematics. The Real Number System Complexity a Complexity b Complexity c
Ohio s Learning Standards-Extended Mathematics The Real Number System Complexity a Complexity b Complexity c Extend the properties of exponents to rational exponents N.RN.1 Explain how the definition of
More informationMA 180 Lecture. Chapter 0. College Algebra and Calculus by Larson/Hodgkins. Fundamental Concepts of Algebra
0.) Real Numbers: Order and Absolute Value Definitions: Set: is a collection of objections in mathematics Real Numbers: set of numbers used in arithmetic MA 80 Lecture Chapter 0 College Algebra and Calculus
More informationIntroduction to Vectors Pg. 279 # 1 6, 8, 9, 10 OR WS 1.1 Sept. 7. Vector Addition Pg. 290 # 3, 4, 6, 7, OR WS 1.2 Sept. 8
UNIT 1 INTRODUCTION TO VECTORS Lesson TOPIC Suggested Work Sept. 5 1.0 Review of Pre-requisite Skills Pg. 273 # 1 9 OR WS 1.0 Fill in Info sheet and get permission sheet signed. Bring in $3 for lesson
More informationVectors. In kinematics, the simplest concept is position, so let s begin with a position vector shown below:
Vectors Extending the concepts of kinematics into two and three dimensions, the idea of a vector becomes very useful. By definition, a vector is a quantity with both a magnitude and a spatial direction.
More informationBSP1153 Mechanics & Thermodynamics. Vector
BSP1153 Mechanics & Thermodynamics by Dr. Farah Hanani bt Zulkifli Faculty of Industrial Sciences & Technology farahhanani@ump.edu.my Chapter Description Expected Outcomes o To understand the concept of
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Two Dimensions; Vectors Vectors and Scalars Units of Chapter 3 Addition of Vectors Graphical Methods Subtraction of Vectors, and Multiplication of a Vector by a Scalar Adding Vectors
More information4/13/2015. I. Vectors and Scalars. II. Addition of Vectors Graphical Methods. a. Addition of Vectors Graphical Methods
I. Vectors and Scalars A vector has magnitude as well as direction. Some vector quantities: displacement, velocity, force, momentum A scalar has only a magnitude. Some scalar quantities: mass, time, temperature
More informationLecture 6: Geometry of OLS Estimation of Linear Regession
Lecture 6: Geometry of OLS Estimation of Linear Regession Xuexin Wang WISE Oct 2013 1 / 22 Matrix Algebra An n m matrix A is a rectangular array that consists of nm elements arranged in n rows and m columns
More informationObjectives: Review open, closed, and mixed intervals, and begin discussion of graphing points in the xyplane. Interval notation
MA 0090 Section 18 - Interval Notation and Graphing Points Objectives: Review open, closed, and mixed intervals, and begin discussion of graphing points in the xyplane. Interval notation Last time, we
More informationFairfield Public Schools
Mathematics Fairfield Public Schools PRE-CALCULUS 40 Pre-Calculus 40 BOE Approved 04/08/2014 1 PRE-CALCULUS 40 Critical Areas of Focus Pre-calculus combines the trigonometric, geometric, and algebraic
More informationTennessee s State Mathematics Standards Precalculus
Tennessee s State Mathematics Standards Precalculus Domain Cluster Standard Number Expressions (N-NE) Represent, interpret, compare, and simplify number expressions 1. Use the laws of exponents and logarithms
More informationIn the real world, objects don t just move back and forth in 1-D! Projectile
Phys 1110, 3-1 CH. 3: Vectors In the real world, objects don t just move back and forth in 1-D In principle, the world is really 3-dimensional (3-D), but in practice, lots of realistic motion is 2-D (like
More informationChapter 2 - Vector Algebra
A spatial vector, or simply vector, is a concept characterized by a magnitude and a direction, and which sums with other vectors according to the Parallelogram Law. A vector can be thought of as an arrow
More informationMathematics for Graphics and Vision
Mathematics for Graphics and Vision Steven Mills March 3, 06 Contents Introduction 5 Scalars 6. Visualising Scalars........................ 6. Operations on Scalars...................... 6.3 A Note on
More information{ independent variable some property or restriction about independent variable } where the vertical line is read such that.
Page 1 of 5 Introduction to Review Materials One key to Algebra success is identifying the type of work necessary to answer a specific question. First you need to identify whether you are dealing with
More informationIntroduction to Matrix Algebra
Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p
More informationMathematics OBJECTIVES FOR ENTRANCE TEST - YEAR 7. Numbers
Mathematics OBJECTIVES FOR ENTRANCE TEST - YEAR 7 1. Adding and subtracting Integers 2. Multiplying and Dividing Integers 3. Adding and Subtracting Decimals 4. Multiplying and Dividing by 10, 100 and 1000
More informationDefinitions In physics we have two types of measurable quantities: vectors and scalars.
1 Definitions In physics we have two types of measurable quantities: vectors and scalars. Scalars: have magnitude (magnitude means size) only Examples of scalar quantities include time, mass, volume, area,
More informationAlgebra I Chapter 6: Solving and Graphing Linear Inequalities
Algebra I Chapter 6: Solving and Graphing Linear Inequalities Jun 10 9:21 AM Chapter 6 Lesson 1 Solve Inequalities Using Addition and Subtraction Vocabulary Words to Review: Inequality Solution of an Inequality
More informationThe geometry of least squares
The geometry of least squares We can think of a vector as a point in space, where the elements of the vector are the coordinates of the point. Consider for example, the following vector s: t = ( 4, 0),
More informationALGEBRA 1. Interactive Notebook Chapter 2: Linear Equations
ALGEBRA 1 Interactive Notebook Chapter 2: Linear Equations 1 TO WRITE AN EQUATION: 1. Identify the unknown (the variable which you are looking to find) 2. Write the sentence as an equation 3. Look for
More informationNewbattle Community High School National 5 Mathematics. Key Facts Q&A
Key Facts Q&A Ways of using this booklet: 1) Write the questions on cards with the answers on the back and test yourself. ) Work with a friend who is also doing National 5 Maths to take turns reading a
More informationExponents. Reteach. Write each expression in exponential form (0.4)
9-1 Exponents You can write a number in exponential form to show repeated multiplication. A number written in exponential form has a base and an exponent. The exponent tells you how many times a number,
More informationVectors. Vector Practice Problems: Odd-numbered problems from
Vectors Vector Practice Problems: Odd-numbered problems from 3.1-3.21 After today, you should be able to: Understand vector notation Use basic trigonometry in order to find the x and y components of a
More informationLecture Notes (Vectors)
Lecture Notes (Vectors) Intro: - up to this point we have learned that physical quantities can be categorized as either scalars or vectors - a vector is a physical quantity that requires the specification
More informationIntroduction to Vectors
Introduction to Vectors Why Vectors? Say you wanted to tell your friend that you re running late and will be there in five minutes. That s precisely enough information for your friend to know when you
More informationMain Ideas in Class Today
Main Ideas in Class Today After today, you should be able to: Understand vector notation Use basic trigonometry in order to find the x and y components of a vector (only right triangles) Add and subtract
More information11.8 Vectors Applications of Trigonometry
00 Applications of Trigonometry.8 Vectors As we have seen numerous times in this book, Mathematics can be used to model and solve real-world problems. For many applications, real numbers suffice; that
More informationUnit Activity Correlations to Common Core State Standards. Precalculus. Table of Contents
Unit Activity Correlations to Common Core State Standards Precalculus Table of Contents Number and Quantity 1 Algebra 3 Functions 3 Geometry 5 Statistics and Probability 6 Number and Quantity The Complex
More informationChapter 8 Scalars and vectors
Chapter 8 Scalars and vectors Heinemann Physics 1 4e Section 8.1 Scalars and vectors Worked example: Try yourself 8.1.1 DESCRIBING VECTORS IN ONE DIMENSION west east + 50 N Describe the vector using: a
More informationHonors Precalculus Yearlong Mathematics Map
Honors Precalculus Yearlong Mathematics Map Resources: Approved from Board of Education Assessments: District Benchmark Assessments Common Core State Standards Standards for Mathematical Practice: 1. Make
More informationExample problem: Free Fall
Example problem: Free Fall A ball is thrown from the top of a building with an initial velocity of 20.0 m/s straight upward, at an initial height of 50.0 m above the ground. The ball just misses the edge
More informationNIOBRARA COUNTY SCHOOL DISTRICT #1 CURRICULUM GUIDE. SUBJECT: Math Trigonometry TIMELINE: 4 th quarter
Expressing Geometric Properties with Equations:Translate between the geometric description and the equation for a conic section Standards: G-GPE.1. Derive the equation of a circle of given center and radius
More informationVectors in Physics. Topics to review:
Vectors in Physics Topics to review: Scalars Versus Vectors The Components of a Vector Adding and Subtracting Vectors Unit Vectors Position, Displacement, Velocity, and Acceleration Vectors Relative Motion
More informationInner Product Spaces 6.1 Length and Dot Product in R n
Inner Product Spaces 6.1 Length and Dot Product in R n Summer 2017 Goals We imitate the concept of length and angle between two vectors in R 2, R 3 to define the same in the n space R n. Main topics are:
More informationREVIEW - Vectors. Vectors. Vector Algebra. Multiplication by a scalar
J. Peraire Dynamics 16.07 Fall 2004 Version 1.1 REVIEW - Vectors By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Since we will making
More informationCHAPTER 2: VECTORS IN 3D 2.1 DEFINITION AND REPRESENTATION OF VECTORS
CHAPTER 2: VECTORS IN 3D 2.1 DEFINITION AND REPRESENTATION OF VECTORS A vector in three dimensions is a quantity that is determined by its magnitude and direction. Vectors are added and multiplied by numbers
More informationMathematics High School Advanced Mathematics Plus Course
Mathematics High School Advanced Mathematics Plus Course, a one credit course, specifies the mathematics that students should study in order to be college and career ready. The Advanced Mathematics Plus
More informationChapter 2. Matrix Arithmetic. Chapter 2
Matrix Arithmetic Matrix Addition and Subtraction Addition and subtraction act element-wise on matrices. In order for the addition/subtraction (A B) to be possible, the two matrices A and B must have the
More informationVectors a vector is a quantity that has both a magnitude (size) and a direction
Vectors In physics, a vector is a quantity that has both a magnitude (size) and a direction. Familiar examples of vectors include velocity, force, and electric field. For any applications beyond one dimension,
More informationFree download from not for resale. Apps 1.1 : Applying trigonometric skills to triangles which do not have a right angle.
Apps 1.1 : Applying trigonometric skills to triangles which do not have a right angle. Area of a triangle using trigonometry. Using the Sine Rule. Using the Cosine Rule to find a side. Using the Cosine
More informationChapter 2. Error Correcting Codes. 2.1 Basic Notions
Chapter 2 Error Correcting Codes The identification number schemes we discussed in the previous chapter give us the ability to determine if an error has been made in recording or transmitting information.
More informationPhysics 170 Lecture 2. Phys 170 Lecture 2 1
Physics 170 Lecture 2 Phys 170 Lecture 2 1 Phys 170 Lecture 2 2 dministrivia Registration issues? Web page issues? On Connect? http://www.physics.ubc.ca/~mattison/courses/phys170 Mastering Engineering
More informationr y The angle theta defines a vector that points from the boat to the top of the cliff where rock breaks off. That angle is given as 30 0
From a boat in the English Channel, you slowly approach the White Cliffs of Dover. You want to know how far you are from the base of the cliff. Then suddenly you see a rock break off from the top and hit
More informationChapter 4. Inequalities
Chapter 4 Inequalities Vannevar Bush, Internet Pioneer 4.1 Inequalities 4. Absolute Value 4.3 Graphing Inequalities with Two Variables Chapter Review Chapter Test 64 Section 4.1 Inequalities Unlike equations,
More information3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Mathematics Standards for Trigonometry/Precalculus The Kansas College and Career Ready high school standards specify the mathematics that all students should study in order to be college and career ready.
More information