Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical statements. Man of the models will be geometric in nature. Thus, we'll need things like the trigonometric relations to establish relations between the components. In the triangle shown here, one angle is marked with a θ. The sides are labeled in relation to this angle: opposite, adjacent, and hpotenuse. So, sin(θ) is defined as the ratio of the side labeled opposite to the side known as the hpotenuse.. The other basic trig functions are defined in similar was: sin(θ) = opp hp cos(θ) = adj hp. and tan(θ) = opp adj. a This one might be a little less familiar, but the same rules appl. b c
Question: Which of the following statements is (are) true? a) sin(α) = cos(β) b) δ = β c) β + γ = ϵ d) tan(α) = tan(γ) Inverse functions We can also use the inverse trigonmetric functions. c a b The inverse trig functions take the ratio of lengths (a dimensionless number) and return an angle (in degrees or radians). opp sin 1 ( hp ) = θ adj cos 1 ( hp ) = θ opp tan 1 ( adj ) = θ Eample Problem: It's 80 meters between 135th and 136th along Edgecombe Ave. Between Edgecombe and St. Nicholas, it's 3 meters. (a) What's the angle between St. Nicholas and Edgecombe? (b) What's the distance between Edgecombe and St. Nicholas along 138th street?
Pthagorean Theorem a b c We'll use this relationship all the time.. a + b = c This is also known as Euclid's 7th proposition from the first book of the Elements. Eample Problem: How far along St. Nicholas is it between 135th and 138th? Scalars and Vectors These are two different mathematical or phsical entities. Scalars: A scalar quantit is completel specified b a single value with an appropriate unit and has no direction. (e.g. $0) Vectors: A vector quantit is completel described b a number and appropriate units plus a direction. (e.g. person walks km E) Some Eamples: Scalars: Temperature, Speed, Distance, length, densit Vectors: Displacement, Velocit, Force, Weight When thinking out the phsical world, ou should intuitivel notice that certain quantities or phenomena have different effects depending on which wa the are pointed, in other words, their effects depend on their direction. For eample, it's a lot easier to walk with the wind blowing in the same direction as our motion, rather than the other wa: walking against the wind. There are two vector quantities at pla in this eample. Your direction of motion (that would be inferred from our velocit vector) and the velocit of the wind. When the point in the same direction, our motion is aided b the wind, when the are in opposite directions, our motion is impeded. Not onl do the magnitudes of these two quantities matter, but so do their directions. And thus, we need to use vectors.
Vector vs. Scalar eample A particle travels from A to B along the path shown b the dotted red line. This is the distance traveled and is a scalar The displacement (change in position) is the solid line from (a) to (b). The displacement is independent of the path taken between the two points displacement is a vector (it has length and direction). a b This image illustrates the difference between displacement and distance traveled. Looking at the dotted line, which represents the distance traveled, compared to the solid displacement vector, we can see that a) its magnitude is probabl much larger than the displacement vector, and b) it doesn't have a clear direction associated with it. Notation When writing math b hand, just put an arrow on top of the variable. This will indicate it is a vector: A In printed tet, ou'll see vectors either in bold face: A or with an arrow: A If we want to refer to the magnitude onl of a vector quantit, we can use absolute value bars: A, or just in italics: A. Wind Map Math properties of vectors Two vectors are equal if the have the same magnitude and the same direction A = B if A = B and the point along parallel lines All of the vectors shown are equal in magnitude and direction, thus the are equal. It might be helpful to think of vectors a notation like this: A = (mag, dir). Thinking this wa we can see that the definition of the vector onl requires two elements, the magnitude and the direction. If we can another vector B, with the same magnitude and direction, it would necessaril have to be equal to A. Addition of Vectors Adding two scalar quantities is eas. We just add them like we would add an normal quantit. However, vectors involve more math. We have to also take into account which wa the are pointing.
Eample Problem: Imagine we walk along two displacement vectors A and B. What is the resultant displacement? Or, what is A + B? Up Edgecombe = 0m in the North direction Along 138th = 50m in the West direction The Resultant C =? Vector Addition: Graphicall To add A + B 1. Arrange the vectors tip to tail.. Connect the tip of A to the origin of B. Link to Vector Addition Sim. Negative of a Vector The negative of a vector is simpl a vector with the same magnitude, but pointed in the opposite direction. The resultant of A + ( A ) = 0 Vector Subtraction To Subtract two vectors, sa A B, all we need to do is add the negative of B to A. Since, A B = A + ( B )
Multiplication of a Vector times a Scalar The result of the multiplication or division b a scalar is a vector. The magnitude of the vector is multiplied or divided b the scalar. Vector Components The components of a vector are the parts of a vector that point along a given ais. We'll use the Cartesian Coordinate Sstem most often. Here we see the and components of the vector A. We can see that the component, A, points all along the ais, while the component, A, points onl along the ais. Here's another vector A decomposed into its and components. Question: Here are the components of R: R = +, = +3 R Which diagram represents R? - - - - - - - -
We can use these vector components to add two arbitrar vectors together. (notice that A and B are not at right angles to each other.) We'll combine the components of A and B to get the components of C. = + C A B = + C A B Once we have the components of C, we can use the pthagorean theorem to get the magnitude of C. C = C + C Eample Problem: (km) A car travels 0 km due N and then 35 km in a direction 60º W of N. Find the magnitude and direction of the car s resultant displacement. N (km) Unit vectors + -z A unit vector is a vector that has a magnitude of eactl 1, and points in a given direction. - + +z - Function of Unit vectors Rather than alwas using the θ and magnitude of a vector to describe it, we can use the unit vectors. A = A i + A j A = 5 i + 3j
Multipling Vectors There are two was of multipling vectors: 1. The dot product (or scalar product) a b = ab cos θ a b = ( a i + a j + a z k ) ( b i + b j + -b z k ) a b = + + a b a b a z b z Multipling Vectors The second method produces another vector:. The cross product (or vector product) a b = ab sin ϕ This produces a third vector that points perpendicular to both the original vectors. - +z +z a b = ( a i + a j + a z k ) ( b i + b j + b z k ) + - + - -z -z + + Question: Which of the following pairs of vectors will have the smallest cross product?
Functional Relations 0 Constant 0 Linear 0 Quadratic Question: g Which of the following functions could describe this plot? ( a is a constant) a) g(h) = a + h b) g(h) = h c) g(h) = a h d) g(h) = h e) g(h) = h 1 0 0 h Simultaneous Solutions Ver often, we'll have two equations and two unknown variables. There are several algebraic tactics that can be emploed to solve the 'sstem'. a + b = a = b
Question: What is the sum of vectors A + B + C + D? a) R b) R c) G d) G e) 0 C B R G A D