Vectors Vectors are a very important type of quantity in science, but what is it? A simple definition might be: Vector: A quantity that has magnitude, unit, and direction; and is commonly represented by a directed line segment whose length represents the magnitude and whose orientation in space represents the direction. That s all fine and well, but what does it mean? 1
What is the difference between a number and a digit? 2
What are Numbers? A number is a mathematical object used to count, label, and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers. Different types of numbers are used in many cases. Numbers can be classified into sets, called number systems. 3
Natural Numbers or Counting Numbers Number systems evolved over time by expanding the notion of what we mean by the word number. At first, number meant something you could count, like how many sheep a farmer owns. These are called the natural numbers, or sometimes the counting numbers. Natural Numbers together with zero ex) 1, 2, 3, 4, 5, 6, 7,... Whole Numbers ex) 0, 1, 2, 3, 4, 5,... Whole numbers plus negatives Integers ex)... 4, 3, 2, 1, 0, 1, 2, 3, 4,... 4
All numbers of the form, zero) A B Rational Numbers where A and B are integers (but B cannot be Rational numbers include all positive and negative fractions, including integers and so called improper fractions. Formally, rational numbers are the set of all real numbers that can be written as a ratio of integers with nonzero denominator. Rational numbers are indicated by the symbol Q. ex) 1.25, 3.3, 8.245, 9.6, 1 2, 7, 15 The bottom of the fraction is called the denominator. Think of it as the denomination it tells you what size fraction we are talking about: fourths, fifths, etc. The top of the fraction is called the numerator. It tells you how many fourths, fifths, or whatever. RESTRICTION: The denominator cannot be zero! (But the numerator can) 3 4 5
Irrational Numbers Irrational numbers don't include integers OR fractions. However, irrational numbers can have a decimal value that continues forever WITHOUT a pattern, unlike rational numbers. An example of a well known irrational number is pi which as we all know is 3.14 but if we look deeper at it, it is actually 3.14159265358979323846264338327950288419...and this goes on for somewhere around 5 trillion digits! π, 2, e 6
7
Digits A digit is a single symbol used to make numbers (numerals). 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are the ten digits we use in everyday numbers. A digit is what is used as a position in place value notation, and a number is one or more digits. Today's most common digits are the decimal (base 10) digits "0", "1", "2", "3", "4", "5", "6", "7", "8", and "9". The distinction between a digit and a number is most pronounced in the context of a number base. Base 10 Number System Base 10 refers to the numbering system in common use. Take a number like 475, base ten refers to the position, the 5 is in the one's place, the 7 is in the ten's place and the 4 is in the hundred's place. Each number is 10 times the value to the right of it, hence the term base ten. The numbers continue indefinitely in this pattern: Hundreds 10 2 100 Tens 10 1 10 Ones 10 0 1... Tenths Notice the middle row, each place value is equal to 10 raised to some power. This type of quantity is called an exponential number. The number follows the form of A n, where A is the base and n is the exponent. In our number system the base is 10 for each place value, which is why it is called the base 10 number system. 10 1 0.1 Hundredths 10 2 0.01 Thousandths 10 3 0.001 2 2 4 2 1 2 Base 2 Number System. 2. 0 2 1. 1 0.5 2 2 0.25 2 3 0.125 8 2 64 8 1 8 Base 8 Number System. 8. 0 8 1. 1 0.125 8 2 0.015625 8 3 0.0019531 8
Base 10 Number System A non zero number with more than one digit position will mean a different number in a different number base, but in general, the digits will mean the same. The base 8 number 23 8 contains two digits, "2" and "3", and with a base number (subscripted) "8", means 19 in base 10. In our notation here, the subscript "8" of the number 23 8 is part of the number, but this may not always be the case. Imagine the number "23" as having an ambiguous base number. Then "23" could likely be any base, base 4 through base 60. In base 4 "23" means 11, and in base 60 it means the number 123. The number "23" then, in this case, corresponds to the set of numbers {11, 13, 15, 17, 19, 21, 23,..., 121, 123} while its digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" means "three of" the value of the place they occupy. Example: The number 586 represents a number that has 5 hundreds, eight tens, and six ones. 9
10
What is the number for the dots in figure 2 below? Figure 2 Note that there is no digit that can be placed in the ones place to indicate this value. Thus, some digit must be placed in the tens place. This digit indicates the number of 10s in the quantity. In figure 2 there is one 10, thus a one is placed in the tens place. When ten dots are removed six are left, thus a six is placed in the ones place. The number is the 16 which represents one ten and six ones, or sixteen. Thus a number is born. Remember, digits are like letters, they are combined to form the number. The number simply represents an amount or a place on a number line. But how does this relate to a vector? numbers are part of the vector quantity. They are the part that indicates how much the quantity represents. This is also known as the magnitude of the vector. 11
Next, it is necessary to associate the number with what amount it was representing. In figure 2 the amount was 16, but 16 of what? Clearly, the quantity is 16 blue dots. Sixteen, being the magnitude, tells how much, and blue dots tells of what. This type of quantity is called a scalar quantity. The part of the quantity which answers the question of what is called the unit. In science a number without a unit has no meaning. It is incorrect to say that the speed of a man who walks 10 meters in 5 seconds is 2. A scientist would ask 2 of what? The correct answer must specify the unit and answer the question of what. Vectors also must contain a unit that answers the question of what. three of what? I'll have three James 12
Finally, vectors need one more thing to fully describe the quantity, a direction. Direction can be as simple as up, down, left, and right. They can be a little harder such as north, south, east, and west. They can also get very precise such as an angle on a polar coordinate plane; 235 o, 53 o, or 326 o. The direction simply answers the question which way that the vector quantity acts. Velocity is a vector quantity. So it would be incorrect to say that the plane had a velocity of 230 miles/hour. This quantity tells how much and of what but not which way, therefore it is NOT a vector and cannot be a velocity. One needs to add a direction such as south to the quantity to make it a vector and a velocity. The quantity 230 miles/hour south is a vector and is a velocity. Therefore, the answer to the question, what is a vector, can be determined. A vector is a quantity that answers three questions. How much? Of What? Which way? 13
Vector Identification The next step in the study of vectors is to learn how to identify vectors. Take for example the dots in figure 2. These dots are a pictorial representation of a scalar quantity. That quantity was simply and quickly identifiable, 16 dots. But vectors are more complex. Since direction is an integral part of the quantity, the pictorial representation of a vector must clearly show the direction. The pictorial representation of a vector is given by a directed line segment. In simplistic terms an arrow, but this arrow must contain 4 distinct elements. These elements are shown below in figure 3. Body Scale 1 cm = 5 m/s Tail (foot) Head 14
Body Scale 1 cm = 5 m/s Tail (foot) Head The tail is the starting point of the vector. The length of the body with the scale represents the magnitude of the vector. (It answers the question how much ) The head shows the direction of the vector. (It answers the question which way ) The scale gives the unit of the vector. (It answers the question of what ) Together they make up the pictorial representation of a vector. But what is its value? Its not as easy as identifying the number of dots in figure 2. 15
Quiz 1) Define Vector: 2) Draw the diagram below and label the parts. B) D) 1 cm = 5 m/s A) C) 1) Define Vector: Vector: A quantity that has magnitude, unit, and direction; and is commonly represented by a directed line segment whose length represents the magnitude and whose orientation in space represents the direction. 2) Draw the diagram below and label the parts. B) Body D) Scale 1 cm = 5 m/s A) Tail C) Head 16
Figure 3 Scale 1 cm = 5 m/s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 0 1 2 3 4 5 The first step in this process is to measure the vector from the tail to the tip of the head. The entire body of the vector needs to be measured. The vector in figure 3 is 8.0 cm long. This length must now be converted from a length of picture unit to the vector unit. This process will also give the magnitude of the vector. Perform this conversion using the factor label method. The scale will be used as the conversion factor. The conversion factor can be seen below: vectors measured length Conversion Factor from scale 8 cm 5 m/s 1 cm = 40 m/s 17
Thus, the magnitude of the vector is 40 and the unit of the vector is m/s (meters per second). However, this is still NOT a vector quantity because it doesn't have a direction. Using a simple north south east and west direction system where the default north position is the top of the paper, a direction can be specified. The direction for the vector in figure 3 is east. Thus, the total vector quantity that the pictorial vector represents is 40 m/s East. N NNE NE ENE W E S 18
Vectors can be directed due East, due West, due South, and due North. But some vectors are directed northeast (at a 45 degree angle); and some vectors are even directed northeast, yet more north than east. Thus, there is a clear need for some form of a convention for identifying the direction of a vector which is not due East, due West, due South, or due North. There are a variety of conventions for describing the direction of any vector. The two conventions which will be discussed and used in this class are described below: A. The direction of a vector is often expressed as an angle relative to some reference point on a coordinate plane. This angle is called theta, θ. This angle is a relative angle that is measured in relation to the closest X axis. (Note: It is perfectly acceptable to measure angles relative to the Y axis, but for the sake of uniformity and ease we will only measure angles relative to an X axis.) Theta, θ is defined as the angle between the vector and the closest X axis. Theta will always be an acute angle between 0 o and 90 o inclusive. B. The direction of a vector is often expressed as an angle relative to some absolute reference point on a coordinate plane. This angle is called phi, φ. This angle is an absolute angle that is measured in relation to the positive X axis. (Note: It is perfectly acceptable to measure angles relative to the any axis, but for the sake of uniformity and ease we will only measure angles relative to the positive X axis.) Phi, φ is defined as the angle between the positive X axis and the vector drawn counter clockwise to the vector. Phi can be any angle between 0 o and 360 o inclusive. 19
Which Vector is the largest? Scale 1 cm = 8 m/s Scale 1 cm = 0.2 m/s Scale 1 cm = 50 m/s Scale 1 cm = 1.5 m/s V 1 V 2 V 3 V 4 7 6 5 2 4 3 11 4 10 9 8 3 7 6 2 5 4 3 1 2 0 180 1 0 2 1 1 90 0 0 20
How to Determine the Scale of a Vector The best choice of scale is one that will result in a diagram that is as large as possible, yet fits on the sheet of paper. The larger the vector picture, is the smaller the drawing and measuring errors will be. For simplicity and uniformity all scales will follow the following format: Scale measurement unit = vector unit With the unit used to measure the vector, usually cm, on the left side of the equality, and the vectors true unit on the right side. Try to set all scales so that 1 cm = some # with the vector unit. Dealing with the conversions becomes much simpler as long as 1 (one) is the factor on cm in the scale. 21
What is the scale used to draw the following vectors? Scale 1 cm = m/s 0 1 2 3 4 5 0 0 1 2 V 1 = 35 m/s East 1) Measure the vector 4 cm 2) Set up a ratio, which is a scale, for the vector where the measurement = the magnitude of the vector. 4 cm = 35 m/s 3) Divide both sides of the equality by the factor multiplying cm on the left of the equal sign. 4 cm = 35 m/s 4 4 1cm = 8.75 m/s Note: As a guideline keep at least 3 digits in all answers 22
Draw a vector that is equal to 3,000 m/s east. Scale 1 cm = m/s 1000 300 600 3000 200 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 0 0 1 2 3 4 5 6 7 8 23
Solve the following: Addition Challenge 1) 3 kg 2) 3 m/s west + 4 kg + 4 m/s north 7 kg 5 m/s north west 24
Vector or Scalar Mass Velocity Acceleration Displacement Time Heat Speed Force Distance Work Energy Momentum Scalar (Magnitude and Unit) Vector (Magnitude, Unit, and Direction) 25
Vector Addition A variety of mathematical operations can be performed with and upon vectors just like they are on scalars. One very important relationship is the addition (and subtraction) of vectors. Vector addition is similar to scalar addition in the following ways: First, vector addition is a binary operation. (Only two vectors can be added at a time.) Second, vector addition is commutative. (The order of addition is unimportant.) Third, when vectors are parallel they behave like numbers on a number line (scalars). One may simply add the magnitudes of vectors that are in the same direction, and subtract the magnitudes of vectors that are in opposite directions. The problems arise when the vectors are not parallel. There are two main methods for adding vectors, graphically, and numerically (mathematically). 26
The graphical methods are equivalent to adding on ones fingers. A first grade student when asked to solve the problem 2 + 3, might hold up two fingers on his left hand and three fingers on his right hand and then count all the fingers. This works as long as the student has enough fingers but how can this be done with vectors. After all, vectors are directionally aware. How is direction specified in a picture? As seen earlier, vectors can be represented by an arrow with a scale. Using this fact will allow vectors to be added graphically. Two methods will be discussed, the Head to Tail method and the Parallelogram method. Head to Tail Method The magnitude and direction of the sum of two or more vectors can be determined by use of an accurately drawn scaled vector diagram. Using a scaled diagram, the head to tail method is employed to determine the vector sum or resultant. A common Physics lab involves a vector walk. Either using centimeter sized displacements upon a map or meter sized displacements in a large open area, a student makes several consecutive displacements beginning from a designated starting position. Suppose that you were given a map of your local area and a set of 18 directions to follow. Starting at home 27
base, these 18 displacement vectors could be added together in consecutive fashion to determine the result of adding the set of 5 directions. Perhaps the first vector is measured 5 m, East. Where this measurement ended, the next measurement would begin. The process would be repeated for all 18 directions. Each time one measurement ended, the next measurement would begin. In essence, you would be using the head to tail method of vector addition. The head to tail method involves drawing a vector to scale on a sheet of paper beginning at a designated starting position. Where the head of this first vector ends, the tail of the second vector begins (thus, head to tail method). The process is repeated for all vectors that are being added. Once all the vectors have been added head to tail, the resultant is then drawn from the tail of the first vector to the head of the last vector; i.e., from start to finish. Once the resultant is drawn, its length can be measured and converted to real units using the given scale. The direction of the resultant can be determined by using a protractor and measuring its counterclockwise angle of rotation from due East. Note with ALL graphical methods there WILL be error due to measuring and drawing!!!! 28
29
Go 30 km north, 60 km west, 40 km south, 30 km east, 20 km north, and 40 km east Scale 1 cm =? Start 0 1 2 3 4 5 6 7 8 9 10 11 0 0 1 2 3 4 30
What is a parallelogram? Which of the following shapes are parallelograms? A) B) C) D) E) 31
Parallelogram Method The parallelogram method involves drawing a vector to scale on a sheet of paper with the tail of each vector starting at the same point. This is the common tail point and can be considered the origin of a coordinate plane. The parallelogram method requires the drawing of parallelograms so one must know something about parallelograms. The parallelogram method of vector addition involves using an accurately drawn, scaled vector diagram to determine the components of the vector. Briefly put, the method involves drawing the vector to scale in the indicated direction, sketching a parallelogram around the vector such that the resultant vector is the diagonal of the parallelogram, and determining the magnitude of the components (the sides of the parallelogram) using the scale. If one desires to determine the components as directed along the traditional x and y coordinate axes, then the parallelogram is a rectangle with sides that stretch vertically and horizontally. 32
Properties of Parallelograms 1) Opposite sides of a parallelogram are parallel (by definition) and equal in length, so will never intersect. 2) The area of a parallelogram is twice the area of a triangle created by one of its diagonals. (The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides.) 3) Any line through the midpoint of a parallelogram bisects the area. 4) The perimeter of a parallelogram is 2(a + b) where a and b are the lengths of adjacent sides. 33
Scale 1 cm = 10 N Parallelogram Method 4 10 9 8 3 7 6 Common tail point 2 5 4 3 1 2 1 90 0 0 X X E 1 Common tail point X R X R = 24 N NNW 34
Pitfall of the Parallelogram Method What happens when 2 vectors are colinear? Adding the Purple and Blue vectors first leads to the situation where the equilibrient vector and the Green vector are colinear! X E 1 X You can't draw a parallelogram when 2 vectors are colinear, therefore the parallelogram method CANNOT be used to add colinear vectors. 35
How to deal with the problem of adding colinear vectors. Method 1: Erase the last parallelogram drawn and use 2 different vectors to form a new parallelogram. X X X E 1 instead x R E 1 x 36
Method 2: Use the head to tail method X E 1 X 1) Find the shorter of the 2 vectors and measure it. 2) Pick the vector "off" the page. X E 1 E 1 X X 3) place the vector back on the page so that the tail of the lifted vector goes on the head of the vector that was left on the page 37
3) place the vector back on the page so that the tail of the lifted vector goes on the head of the vector that was left on the page E 1 X E 1 X X X E 1 R E 1 X X 4) The resultant vector connects the tail of the vector left of the page to the head of the vector that was lifted off the page. 38
x R E 1 x X E 1 R E 1 X X 39
Numerical Vector Addition Numerical vector addition, sometimes called the trigonometric method or the component method, is a non graphical method for adding vectors. Therefore it does not have any inherent drawing errors in the resolution of the resultant vector. The method requires that the vectors be broken down into their component parts. These parts can then be combined using trigonometry to determine the resultant vector. Because numerical vector addition is requires trigonometry, We will look at some basic properties of trigonometry. 40
Trigonometry Trigonometry began as the computational component of geometry. For instance, one statement of plane geometry states that a triangle is determined by a side and two angles. In other words, given one side of a triangle and two angles in the triangle, then the other two sides and the remaining angle are determined. Trigonometry includes the methods for computing those other two sides. The remaining angle is easy to find since the sum of the three angles equals 180 degrees (usually written 180 ). If there is anything that distinguishes trigonometry from the rest of geometry, it is that trig depends on angle measurement and quantities determined by the measure of an angle. Of course, all of geometry depends on treating angles as quantities, but in the rest of geometry, angles aren't measured, they're just compared or added or subtracted. 41
Naming the parts of a right triangle β Hypotenuse The side opposite the right angle is the hypotenuse α β Hypotenuse Pick a reference angle either α or β. For this example, let α be the reference angle. α 42
β Hypotenuse The side that forms the reference angle with the hypotenuse is called the adjacent side. α Adjacent Side Reference Angle (Adjacent meaning "next to") Opposite Side β Hypotenuse α Adjacent Side The side across from the reference angle is called the opposite side. Reference Angle 43
If the reference angle is changed to β, then the adjacent side and opposite side are determined relative to the new angle. Note: the hypotenuse does NOT change! Adjacent Side β Reference Angle Hypotenuse α Opposite Side 44
A quick and easy way to find the opposite side is to think of Pacman. α Opposite Side Think of Pacman's mouth as forming the reference angle (α) in the triangle (It doesn't have to be a right triangle). Pacman will then eat the opposite side of the triangle. 45
The Pythagorean Theorem Before embarking on trigonometry, there are a couple of things you need to know well about geometry, namely the Pythagorean theorem and similar triangles. Both of these are used over and over in trigonometry. The Pythagorean theorem is about right triangles, that is, triangles, one of whose angles is a 90 angle. A right triangle is displayed in the diagram below. Leg right angle hypotenuse Opposite Side right angle hypotenuse θ Reference angle Leg Adjacent Side (Hypotenuse) 2 = (Adjacent Side) 2 + (Opposite Side) 2 46
Similar triangles Two triangles ABC and DEF are similar if (1) their corresponding angles are equal, that is, angle A equals angle D, angle B equals angle E, and angle C equals angle F, and (2) their sides are proportional, that is, the ratios of the three corresponding sides are equal: B F A C E D AB DE = BC EF = CA FD 47
The Interior Angles of a Triangle add up to 180 This works for ALL triangles Ʃ intierior angles = 180 β β = 30 β β = 50 90 60 110 20 48
Sine The sine of an angle in a right triangle equals the opposite side divided by the hypotenuse: Opposite Side β Hypotenuse α Adjacent Side Sin α = Reference Angle Opp Side Hypotenuse 49
Cosine The cosine of an angle in a right triangle equals the adjacent side divided by the hypotenuse: Opposite Side β Hypotenuse α Adjacent Side Cos α = Reference Angle Adj Side Hypotenuse 50
Tangent The tangent of an angle in a right triangle equals the opposite side divided by the adjacent side: Opposite Side β Hypotenuse α Adjacent Side Tan α = Tan α = Reference Angle Opp Side Adj Side Sin α Cos α 51
A mnemonic to help remember these Trigonometric functions is: S O H C A H T O A sine Opposite Hypotenuse cosine Adjacent Hypotenuse Tangent Opposite Adjacent (Hypotenuse) 2 = (Adjacent Side) 2 + (Opposite Side) 2 Ʃ intierior angles = 180 52
Find angle β, and sides A and B. A β 12 First, specify the reference angle and name the sides of the triangle. Second, determine the trigonometric relationship that solves for the unknowns. 30 o B 53
β Hypotenuse Opposite Side A 12 30 o Reference Angle B Adjacent Side 1) Find angle β using the relationsip that the sum of the interior angles is equal to 180 o. 90 o + 30 o + β = 180 o β = 60 o 2) Find side B. Side B is the adjacent side, the reference angle is 30 o, and the hypotenuse is known ( H = 12). The trigonometric function that contain all these quantities is cosine. Cos β = Adj Side Hypotenuse Cos 30 o = B 12 B = 12 Cos 30 o B = 10.39 3) Find side A. Side A is the opposite side, the reference angle is 30 o, and the hypotenuse is known ( H = 12). The trigonometric function that contain all these quantities is sine. Opp Side Sin β = Hypotenuse Sin 30 o = A 12 A = 12 Sin 30 o A = 6 54
C β A Find the unknowns C 20 γ 26 o 18 o A 8 Finding β Ʃ intierior angles = 180 180 = 26 o + 90 o + β 180 = 116 o + β 64 = β Finding C Knowns: Reference angle 26 o and adjacent side 8 Side C is the opposite side Tan α = Trig Function Tan 26 o = Opp Side Adj Side C 8 C = (8)(Tan 26 o ) C = 3.9 Finding A Knowns: Reference angle 26 o and adjacent side 8 Side A is the Hypotenuse Cos α = Trig Function Cos 26 o = A = Adj Side Hypotenuse A = 8.9 8 8 A Cos 26 o A = Finding γ Ʃ intierior angles = 180 180 = 18 o + 90 o + γ 180 = 108 o + γ 72 = γ Finding C Knowns: Reference angle 18 o and opposite side 20 Side C is the adjacent side Trig Function Tan α = Tan 18 o = C = Opp Side Adj Side 20 20 Sin 18 o A = 64.7 20 C Tan 18 o C = 61.6 Finding A Knowns: Reference angle 18 o and opposite side 20 Side A is the Hypotenuse Trig Function Sin α = Sin 18 o = Opp Side Hypotenuse 20 A 55
C β 30 Find the unknowns C A γ 33 o 22 o 20 B β = 57 o B = 25.2 C = 16.3 γ = 68 o A = 7.5 C = 18.5 56
Find the unknowns β 6 Q (not from Bond movies) α 8 57
In order to solve the problem on the previous page, we must use the arc function or inverse function, but what is an arc function? In mathematics, the arc or inverse trigonometric functions are the inverse functions of the trigonometric functions with suitably restricted domains. (?) The notations sin 1, cos 1, tan 1, etc. are often used for arcsin, arccos, arctan, etc., but this convention logically conflicts with the common semantics for expressions like sin 2 (x), which refer to numeric power rather than function composition, and therefore may result in confusion between multiplicative inverse and compositional inverse. Note: sin 1 1 sin In order to explain this we must first understand what a function is. A function is a relation for which each value from the set of first components of the ordered pairs is associated with exactly one value from the set of second components of the ordered pair. The function explains how the input value (independent variable) will be paired with the output value (dependent variable). Simply stated it's the math that is performed on the input value to get the output value. 58
Consider the following function f(x) = 5x + 8 The function states to multiply the input value by 5 and then add 8. X is the input variable, its value does not depend on the function. Thus, the value of X is independent of the function and is called the independent variable. The function f(x) is sometimes assigned a variable such as y, where f(x) = y The output value of the function (y) depends on the type of function and the input variable. Since its value depends on the function it is called the dependent variable. Evaluate the function by determining the output variable based on several input values. X 3 2 1 0 1 2 3 f(x) = y f(x) = 5x + 8 This is the normal process when evaluating functions. But what happens if the output value is known, is it possible to determine the input variable? X f(x) = y 58 22 123 The calculation of the input value from a known output is called the inverse functionality. It is the inverse of what was done in the table to the right. When this is applied to trigonometric functions, we call them arc functions. 59
Law of Sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles. According to the law, a b c Sin α = Sin β = Sin γ where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see the figure to the left). (sometimes the law is stated using the reciprocal) The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known a technique known as triangulation. It can also be used when two sides and one of the non enclosed angles are known. 60
When to use the law of sines formula You should use the law of sines when you know 2 sides and an angle (case 1 in the picture below) and you want to find the measure of an angle opposite a known side. Or when you know 2 angles and 1 side and want to get the side opposite a known angle (case 2 in picture below). In both cases, you must already know a side and an angle that are opposite of each other. Case 1 2 Sides 1 Angle Case 2 1 Side 2 Angles β 49 c 118 11 γ c 29 a 115 γ 16 Note: 115 is a non enclosed angle while γ is an enclosed angle. 61
Law of Cosines In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a plane triangle to the cosine of one of its angles. The law of cosines says c 2 = a 2 + b 2 2ab cos γ a 2 = c 2 + b 2 2cb cos α b 2 = a 2 + c 2 2ac cos β Note: the side to the left of the equal sign and the angle with cosine are ALWAYS opposite each other in the triangle. 62
The law of cosines is a formula that relates the three sides of a triangle to the cosine of a given angle. This formula allows you Case I) to calculate the side length of non right triangle as long as you know two sides and an angle Case 2) to calculate any angle of a triangle if you know all three side lengths Case 1 Case 2 52 115 16 32 24 A 37 63
1) Using the Law of Sines and the Law of cosines 2) c β 49 c 118 33 γ 115 16 γ 29 a 64
3) 4) α 11 β 37 8 5 α γ 8 9 β c 65
Trig Extra cos (90 θ) = sin θ θ 90 θ Ycomp sin (90 θ) = cos θ Ycomp θ Xcomp Xcomp Xcomp = V sin θ Ycomp = V cos θ V Xcomp 90 θ Ycomp Xcomp = V cos 90 θ Ycomp = V sin 90 θ Since: Xcomp = Xcomp V sin θ = V cos 90 θ sin θ = cos 90 θ Ycomp = Ycomp V cos θ = V sin 90 θ cos θ = sin 90 θ 66
S O H C A H T O A sine Opposite Hypotenuse cosine Adjacent Hypotenuse Tangent Opposite Adjacent (Hypotenuse) 2 = (Adjacent Side) 2 + (Opposite Side) 2 Ʃ intierior angles = 180 Law of Sines a b c Sin α = Sin β = Sin γ c 2 = a 2 + b 2 2ab cos γ Law of Cosines a 2 = c 2 + b 2 2cb cos α b 2 = a 2 + c 2 2ac cos β 67
1 β A 32 o Quiz 2 C 25 C 15 28 o A γ gamma 3 4 C β 115 49 16 γ gamma 11 α β C 28 Find side C only 7.6 68
Quiz Key 1 2 β = 58 o A = 17.7 C = 9.4 3 4 β = 17.2 o γ = 47.8 o C = 40.1 γ = 62 o A = 53.3 C = 47.0 C = 5.6 α = 39.6 o β = 112.4 o 69
Lab Measuring Height Indirectly using Trigonometry Purpose: To develop a method for measuring the height of a flagpole using trigonometry. Materials: small piece of string, nut, straw, protractor, 50 cm meter stick, 100 cm meter stick. Using ONLY the materials stated above develop a procedure for measuring the height of a flagpole. 70
Vector Components Any vector directed in two dimensions can be thought of as having an influence in two different directions. That is, it can be thought of as having two parts. Each part of a two dimensional vector is known as a component. The components of a vector depict the influence of that vector in a given direction. The combined influence of the two components is equivalent to the influence of the single two dimensional vector. The single twodimensional vector could be replaced by the two components. This is true for scalars as well. Think of the scalar quantity 9 dots. We can break the quantity up into 2 parts, say 3 dots and 6 dots. The combined influence of the two parts is equivalent to the original quantity. See below: 71
Vector Components Consider Fido below, if Fido's dog chain is stretched upward and rightward and pulled tight by his master, then the tension force in the chain has two components an upward component and a rightward component. To Fido, the influence of the chain on his body is equivalent to the influence of two chains on his body one pulling upward and the other pulling rightward. If the single chain were replaced by two chains. with each chain having the magnitude and direction of the components, then Fido would not know the difference. This is not because Fido is dumb, but rather because the combined influence of the two components is equivalent to the influence of the single two dimensional vector. 72
Vector Components To determine vector components, we need to find a system where the two dimensional vector can be broken up into 2 pieces (vectors in 3 dimensions are broken into 3 parts). This system has to be uniform, so that everyone is referencing the same thing. The agreed upon convention is to break the vector this way: 1. Place the vector on a Cartesian Coordinate plane. 2. Place the tail of the vector at the origin. 3. Break the vector into pieces relative to the X and Y directions (for 2 dimensions); and X, Y, and Z directions (for 3 dimensions). 4. Use trigonometry to determine the value for the components. 5. The direction of the component vector will correspond to the direction of the unit vector. 73
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0,0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. axis 74
Unit Vector a unit vector in a normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length). A unit vector is often denoted by a lowercase letter with a "hat", like this: v ^ (pronounced "v hat") 2 Dimensions + Y 3 Dimensions + j X i + i + X j Y 75
Vector Components Now that we know some trigonometry, how does this apply to vectors. A vector can be named by a single letter, such as v. The vector v is symbolized by a letter v with an arrow above it, like this: v. A vector is determined by two coordinates, just like a point one for its magnitude in the x direction, and one for its magnitude in the y direction. The magnitude of a vector in the x direction is called the horizontal, or x component of the vector. The magnitude of a vector in the y direction is called the vertical, or the y component of the vector. A vector v with coordinates (3,4) and origin at the origin of the cartesian coordinate plane looks like this: We calculate the the value of the X and Y components using trigonometry 76
Vector Components Start by placing the vector on a Cartesian coordinate plane. The origin of the coordinate plane is the common tail point for all the vectors. V 1 Vector V 1 is placed on the coordinate plane so that its tail starts at the origin. x axis θ axis y axis 77
Vector Components Find the X component of the vector. The X component is the amount of the vector acting in the X direction. Graphically the X component can be determined by: x axis θ X Comp V 1 1. Place the vector on the coordinate plane. 2. imagine a light shines down on the X axis. 3. the shadow cast on the X axis is the X component. 4. The length of the shadow is its magnitude 5. The direction is the direction of the unit vector. axis y axis 78
Vector Components Find the Y component of the vector. The Y component is the amount of the vector acting in the Y direction. Graphically the Y component can be determined by: x axis Y Comp θ V 1 1. Place the vector on the coordinate plane. 2. imagine a light shines down on the Y axis. 3. the shadow cast on the Y axis is the Y component. 4. The length of the shadow is its magnitude 5. The direction is the direction of the unit vector. axis y axis 79
Vector Components Next, take the X comp and the Y comp and place the head to tail. the resultant vector is the original vector. V 1 at θ θ Y comp Notice this forms a right triangle where The original vector is the hypotenuse. θ is the reference angle. The X comp is the adjacent side. The Y comp is the opposite side. X comp When the original vector is given, we can use our knowledge of trigonometry to determine the X comp and the Y comp. 80
Vector Components The X comp is found using the cosine function: V 1 at θ θ Y comp Cos α = Adj Side Hypotenuse Note: the hypotenuse is the magnitude of the given vector V 1. X comp Substituting gives the equation below: Cos θ = X comp V 1 Solving for the X comp X comp = V 1 Cos θ This is the equation to find X components V 1 is the Magnitude of the original vector. 81
Vector Components The Y comp is found using the sine function: V 1 at θ θ X comp Y comp Sin α = Opp Side Hypotenuse Note: the hypotenuse is the magnitude of the given vector V 1. Substituting gives the equation below: Sin θ = Solving for the Y comp Y comp V 1 Y comp = V 1 Sin θ This is the equation to find Y components V 1 is the Magnitude of the original vector. 82
Vector Components Find the X and Y components for the following vectors: V 1 = 10 N at 60 o V 2 = 10 N at 60 o V 3 = 10 N at 60 o V 4 = 10 N at 60 o 60 o 60 o 60 o 60 o Xcomp = Xcomp = Xcomp = Xcomp = Ycomp = Ycomp = Ycomp = Ycomp = 83
Recall the Definition of the angles θ and Φ A. The direction of a vector is often expressed as an angle relative to some reference point on a coordinate plane. This angle is called theta, θ. This angle is a relative angle that is measured in relation to the closest X axis. (Note: It is perfectly acceptable to measure angles relative to the Y axis, but for the sake of uniformity and ease we will only measure angles relative to an X axis.) Theta, θ is defined as the angle between the vector and the closest X axis. Theta will always be an acute angle between 0 o and 90 o inclusive. B. The direction of a vector is often expressed as an angle relative to some absolute reference point on a coordinate plane. This angle is called phi, φ. This angle is an absolute angle that is measured in relation to the positive X axis. (Note: It is perfectly acceptable to measure angles relative to the any axis, but for the sake of uniformity and ease we will only measure angles relative to the positive X axis.) Phi, φ is defined as the angle between the positive X axis and the vector drawn counter clockwise to the vector. Phi can be any angle between 0 o and 360 o inclusive. 84
Define the Following: Theta, θ: The angle between the vector and the closest X axis. Theta will always be an acute angle between 0 o and 90 o inclusive. Phi, φ: The angle between the positive X axis and the vector drawn counterclockwise to the vector. Phi can be any angle between 0 o and 360 o inclusive. 85
II θ and Φ I V 1 Quad I Φ = θ Xcomp = + Ycomp = + X axis θ Φ 0 o < Φ < 90 o axis III Y axis IV 86
II θ and Φ I Quad II Φ = 180 o θ V 2 Xcomp = Ycomp = + Φ 90 o < Φ < 180 o X axis θ axis III Y axis IV 87
II θ and Φ I Φ X axis Quad III Φ = 180 o + θ θ axis Xcomp = Ycomp = 180 o < Φ < 270 o V 3 III Y axis IV 88
II θ and Φ I X axis Φ θ axis Quad IV Φ = 360 o θ Xcomp = + Ycomp = III Y axis IV V 4 270 o < Φ < 360 o 89
II θ and Φ θ = 90 o Φ = 90 o I Quad II Φ = 180 o θ Xcomp = Ycomp = + θ = 0 o X axis Φ = 180 o V 2 Φ θ θ Φ θ Φ θ V 1 Quad I Φ = θ Xcomp = + Ycomp = + axis θ = 0o Φ = 0 o Quad III Φ = 180 o + θ Xcomp = Ycomp = V 3 V 4 Φ Quad IV Φ = 360 o θ Xcomp = + Ycomp = III θ = 90 o Y axis Φ = 270 o IV 90
Vector Components Find the X and Y components for the following vectors using angle Φ: V 1 = 10 N at 60 o V 2 = 10 N at 60 o V 3 = 10 N at 60 o V 4 = 10 N at 60 o 60 o 60 o 60 o 60 o θ = θ = θ = θ = Φ = Φ = Φ = Φ = Xcomp = Xcomp = Xcomp = Xcomp = Ycomp = Ycomp = Ycomp = Ycomp = 91
θ theta Φ phi Xcomp = V 1 cos θ or Xcomp = V 1 cos Φ Ycomp = V 1 sin θ or Ycomp = V 1 sin Φ Using θ will always give a positive component. Therefore, you must determine the correct sign yourself. Using Φ will always give the correct sign on the component. 92
Vector Components Find angles θ and Φ; and the X and Y components for the following vectors: V 4 = 10 N at 142 o V 6 = 10 N at 72 o 318 o V 5 = 10 N at 318 o 72 o V 7 = 10 N at 53 o 53 o 142 o θ = θ = θ = θ = Φ = Φ = Φ = Φ = Xcomp = Xcomp = Xcomp = Xcomp = Ycomp = Ycomp = Ycomp = Ycomp = 93
Xcomp = V 1 cos θ II θ and Φ θ = 90 o Φ = 90 o I Ycomp = V 1 sin θ Quad II Φ = 180 o θ Xcomp = Ycomp = + θ = 0 X o axis Φ = 180 o V 2 Φ θ θ Φ θ Φ θ V 1 Quad I Φ = θ Xcomp = + Ycomp = + axis θ = 0o Φ = 0 o Quad III Φ = 180 o + θ Xcomp = Ycomp = V 3 V 4 Φ Quad IV Φ = 360 o θ Xcomp = + Ycomp = III Y axis θ = 90 o Φ = 270 o IV 94
Colinear Vector Addition When vectors are colinear vector addition is analogous to scalar addition. The sign on the vector coorosponds to the direction of the vector: positive when the angle is 0 o and negative when the angle is 180 o. 3 N at 0 o 2 N at 0 o 5 N at 0 o 3 N + 2 N = 5 N 3 N at 0 o 2 N at 180 o 1 N at 0 o 3 N + 2 N = 1 N 95
Colinear Vector Subtraction When vectors are colinear vector subtraction is accomplished by changing the direction of the vector that is being subtracted.,,,,,,,,,,, 3 N at 0 o 2 N at 0 o Since 2 N at 0 o 2 N at 180 o The equation becomes; 3 N at 0 o 2 N at 180 o 1 N at 0 o 3 N + 2 N = 1 N 96
Numerical Vector Addition Now that there is a system to get component vectors to be colinear, the addition of vectors numerically reduces to several simple steps. The steps are outlined in the following example. Add the following 3 vectors. 20 N 10 N 40 o 30 o 45 o Note: The diagrams are NOT drawn to scale. 40 N 97
Numerical Vector Addition 20 N Step 1: Find the X and Y components for each vector to be added. X = 15.3 N Y = 12.9 N 40 o 30 o 10 N X = 8.7 N Y = 5.0 N 45 o Make Sure the components have the proper signs. 40 N X = 28.3 N Y = 28.3 N 98
Numerical Vector Addition 20 N X = 15.3 N Y = 12.9 N 40 o 30 o 10 N X = 8.7 N Y = 5.0 N Step 2: Add all the X components together (make sure they are added with the proper signs!). This is the X component of the resultant vector Rx. 45 o Rx = Ʃ X components Rx = 8.7 N + ( 15.3 N) + ( 28.3 N) 40 N Rx = 34.9 N X = 28.3 N Y = 28.3 N 99
Numerical Vector Addition 20 N X = 15.3 N Y = 12.9 N 40 o 30 o 10 N X = 8.7 N Y = 5.0 N Step 3: Add all the Y components together (make sure they are added with the proper signs!). This is the Y component of the resultant vector Ry. 45 o Ry = Ʃ Y components Ry = 5.0 N + 12.9 N + ( 28.3 N) 40 N Ry = 10.4 N X = 28.3 N Y = 28.3 N 100
Numerical Vector Addition 20 N X = 15.3 N Y = 12.9 N 40 N 40 o 45 o X = 28.3 N Y = 28.3 N 30 o 10 N X = 8.7 N Y = 5.0 N Rx = 34.9 N Ry = 10.4 N Step 4: Find the magnitude of the resultant vector. Rx is the X comp of the resultant and Ry is the Y comp of the resultant. If we add Rx and Ry, we should get the resultant. Because Rx and Ry are always on the X and Y axis, they will always be perpendicular to each other. Thus, the vector addition will always form a right triangle. Using the head to tail method, Rx and Ry are added below: Ry Rx R θ 101
20 N X = 15.3 N Y = 12.9 N 40 N 40 o 45 o X = 28.3 N Y = 28.3 N 30 o Rx = 34.9 N Ry = 10.4 N Numerical Vector Addition 10 N X = 8.7 N Y = 5.0 N Step 4: Find the magnitude of the resultant vector. Ry = 10.4 N Rx = 34.9 N Notice R is the hypotenuse of the right triangle. We can use the Pythagorean Theorem to calculate R. R θ R 2 = Rx 2 + Ry 2 R = Rx 2 + Ry 2 102
Numerical Vector Addition 20 N X = 15.3 N Y = 12.9 N 40 o 30 o 10 N X = 8.7 N Y = 5.0 N Step 4: Find the magnitude of the resultant vector. Use the Pythagorean Theorem to find the magnitude of the resultant vector R. 40 N 45 o X = 28.3 N Y = 28.3 N Rx = 34.9 N Ry = 10.4 N (Notice the negative signs are squared) R = Rx 2 + Ry 2 R = ( 34.9 N) 2 + ( 10.4 N) 2 R = 1218.01 N 2 + 108.16 N 2 R = 1326.17 N 2 R = 36.4 N Note: This is NOT the final answer, it is only the magnitude of the final answer. We still need to calculate direction. 103
Numerical Vector Addition 20 N X = 15.3 N Y = 12.9 N 40 N 40 o 45 o X = 28.3 N Y = 28.3 N 30 o 10 N X = 8.7 N Y = 5.0 N Rx = 34.9 N Ry = 10.4 N R = 36.4 N Step 5: Find the angle θ for the resultant vector. Again look at the triangle formed by Rx, Ry, and R below. Rx Ry R θ is the reference angle, making Rx the adjacent side and Ry the opposite side. We can use the Tangent function to calculate θ. Tan θ = Opp Side Adj Side Ry Solving for θ yields θ = Tan ( 1 Rx) θ = Ry Rx 104
Numerical Vector Addition 20 N X = 15.3 N Y = 12.9 N 40 N 40 o 45 o X = 28.3 N Y = 28.3 N 30 o 10 N X = 8.7 N Y = 5.0 N Rx = 34.9 N Ry = 10.4 N R = 36.4 N Step 5: Find the angle θ for the resultant vector using the arc tangent equation Ry θ = Tan ( 1 Rx) Note: Angle θ must be positive which is why we have the absolute value bars. θ = Tan 1 ( ) 10.4 N 34.9 N θ = Tan 1 ( 0.29799426934) θ = 16.6 o Note: this is NOT the final direction, we must find the absolute angle Φ. 105
Numerical Vector Addition 20 N X = 15.3 N Y = 12.9 N 40 N 40 o 45 o X = 28.3 N Y = 28.3 N 30 o 10 N X = 8.7 N Y = 5.0 N Rx = 34.9 N Ry = 10.4 N R = 36.4 N θ = 16.6 o Step 6: Find the angle Φ for the resultant vector using the proper phi formula. To determine the proper phi formula to use determine which quadrent the resultant vector is pointing in. To determine the quadrent, look at the signs for Rx and Ry. The signs will indicate which quadrent the result vector is in. Rx = Ry = + Rx = + Ry = + Rx = Ry = Rx = + Ry = 106
Numerical Vector Addition 20 N X = 15.3 N Y = 12.9 N 40 N 40 o 45 o X = 28.3 N Y = 28.3 N 30 o 10 N X = 8.7 N Y = 5.0 N Rx = 34.9 N Ry = 10.4 N R = 36.4 N θ = 16.6 o Step 6: Find the angle Φ for the resultant vector using the proper phi formula. For this example Rx is negative and Ry is negative, which indicates that the resultant vector is in quadrent III. The quadrent III phi formula is Φ = 180 o + θ Φ = 180 o + 16.6 o Φ = 196.6 o 107
Numerical Vector Addition 20 N X = 15.3 N Y = 12.9 N 40 N 40 o 45 o X = 28.3 N Y = 28.3 N 30 o 10 N X = 8.7 N Y = 5.0 N Rx = 34.9 N Ry = 10.4 N R = 36.4 N θ = 16.6 o Φ = 196.6 o The Final answer will follow the following format: R = R at Φ Where R is the resultant vector. R is the magnitude of the resultant vector found in step 4. Φ is the absolute angle found in step 6. R = 36.4 N at 196.6 o 108
Step 1: Find the X and Y components for each vector to be added. (Xcomp = V 1 cos θ Ycomp = V 1 sin θ) Step 2: Add all the X components together (make sure they are added with the proper signs!). This is the X component of the resultant vector Rx. Step 3: Add all the Y components together (make sure they are added with the proper signs!). This is the Y component of the resultant vector Ry. Step 4: Find the magnitude of the resultant vector. Use the Pythagorean Theorem to find the magnitude of the resultant vector R. R = Rx 2 + Ry 2 Step 5: Find the angle θ for the resultant vector using the arc tangent equation. Ry θ = Tan ( 1 Rx) Step 6: Find the angle Φ for the resultant vector using the proper phi formula. 109
1 25 N Numerical Vector Addition Solve the Following 34 o 42 o 15 N 30 N 12 N 110
2 25 N Numerical Vector Addition Solve the Following 22 N 42 o 34 o 45 N 55 N 111
3 62 N Numerical Vector Addition 22 N Solve the Following 52 o 22 o 62 o 44 o 28 N 15 N 112
4 20 N Numerical Vector Addition 50 N Solve the Following 30 o 60 o 62.4 o 45 o 30 N 40 N 113
5 60 N Numerical Vector Addition 26.3 N Solve the Following 35 o 46 o 32 o 33 o 58 N 80 N 114
6 27 N Numerical Vector Addition Solve the Following 42.1 N 25 o 55.37 o 47 o 38 o 36 N 32 N 115
7 66 N Numerical Vector Addition 47 N Solve the Following 55 N 54 o 55 o 100 o 39 o 88 N 77 N 116
117