Intro to Phys Sci I Mechanics Heat Instructor Carlos Perez Vector and Scalar Quantities 1 Scalar Quantities When you want to know the temperature outside so that you will know how to dress the only information you need is a number and the unit degrees or degrees F Temperature is therefore an example of a scalar quantity A scalar quantity is completely speci ed by a single value with an appropriate unit and has no direction For example a mass of 2 kg In general scalar quantities can be positive or negative Some scalar physical quantities only take positive values For example Time 5 seconds 3 minutes We don t consider negative times Mass 905 grams 13 56 kilograms Volume 355 mL 1 5 gallons 12 oz uid ounces Temperature is an example of a scalar quantity that can take both positive and negative values For instance we can report temperatures such as 5 C Of course there are positive temperatures like 73 F The operations that we can perform with scalar quantities are the same we already know addition subtraction multiplication division Remember to always make sure that the units are consistent 3 m s 2 s 1 5 m 3 cid 16 m cid 17 1 5 3 m s 3 2 s 6 m 0 5 m 5 5 m 2 m 2 3 m 3 4 m2 27 m3 NONSENSE 2 Vector Quantities If you are preparing to pilot a small plane and need to know the wind velocity you must know both the speed of the wind and its direction Because direction is important for its complete speci cation velocity is a vector quantity A vector quantity is completely speci ed by a number with an appropriate unit the magnitude of the vector plus a direction The magnitude is the how much or how big part This number is never negative To understand more about vectors and how they combine we start with the simplest vector quantity displacement Displacement is simply a change in the position of an object Displacement is a vector quantity because we must state not only how far the object moves but also in what direction Walking 3 km north from your front door doesn t get you to the same place as walking 3 km southeast these two displacements have the same magnitude but di erent directions We always draw a vector as a line with an arrowhead at its tip The length of the line shows the vector s magnitude and the direction of the line shows the vector s direction Symbolically and especially for mathematical notation we represent a vector quantity by a single letter with an arrow above it cid 126 A cid 126 x cid 126 S etc 1 Figure 1 Representing vector quantities with arrows Some other physical quantities that are vector quantities forces acceleration momentum velocity electric eld magnetic eld 3 Cartesian Coordinate Systems and Trigonometry We will need to remember what a Cartesian or rectangular coordinate system is In this system perperdicular axes intersect at a point de ned as the origin O Figure 2 Designation of points in a Cartesian coordinate system Every point is labeled with coordinates x y Sometimes it is more convenient to represent a point in a plane by its plane polar coordinates r as shown in Figure a In this polar coordinate system r is the distance from the origin to the point having Cartesian coordinates x y and is the angle between a xed axis and a line drawn from the origin to the point The xed axis is often the positive x axis and is usually measured counterclockwise from it From the right triangle in Figure b we nd that sin and that cos y r x r Figure 3 a The plane polar coordinates of a point are represented by the distance r and the angle where is measured counterclockwise form the positive x axis b The right triangle used to relate x y to r 2 Therefore starting with the plane polar coordinates of any point we can obtain the Cartesian coordinates by using the equations Furthermore if we know the Cartesian coordinates the de nitions of trigonometry tell us that x r cos y r sin cid 16 y cid 17 x tan y x arctan r cid 112 x2 y2 1 2 3 4 Notice that the equation 4 is the Pythagorean theorem NOTE It is important to maintain the convention for measuring angles There are four regions or quadrants in a Cartesian coordinate system as seen in the gure below Figure 4 The four quadrants in a Cartesian plane The numbering follows a counterclockwise sense The positive x axis corresponds to an angle of 0 the positive y axis corresponds to an angle of 90 the negative x axis corresponds to an angle of 180 and the negative y axis corresponds to an angle of 270 The following table summarizes the sign convention and intervals for angles by quadrant Quadrant Sign of x coordinate Sign of y coordinate Angle degrees First Second Third Fourth Positive Negative Negative Positive Positive Positive Negative Positive 0 90 90 180 180 270 270 360 Example Let s look at the following triangle To obtain the corresponding coordinates correctly we need to determine the angle measured counterclockwise from the positive x axis That doesn t correspond with the angle in the diagram below 3 However we know that the angle we are looking for say plus the angle in the diagram equals 270 Therefore Figure 5 Determining the correct angle to use 35 270 270 35 235 Before we proceed with any calculations let s use some intuition From the picture we expect both the x and y coordinates to be negative Let s see if this happens x 5 cos 235 3 44 y 5 sin 235 4 91 We automatically get the right signs for both coordinates No need to put them by hand Thus the importance of using the correct angle 4 How to Represent and Write Vector Quantities Now that we have a better idea of what a scalar and a vector is we need a way to write vector quantities such that we can perform mathematical operations with them We can draw arrows that represent vectors in up to three dimensions like the real world that has three spatial dimensions We can also draw arrows in two dimensions where this description is accomplished with the use of the Cartesian or rectangular coordinate system We will learn about two ways to write a vector quantity mathematically the magnitude and direction or polar form and the component form closely related to coordinates of a Cartesian system 4 1 Magnitude and direction Form One way to characterize or describe a vector mathematically is by writing its magnitude and its direction When drawing arrows and diagrams the magnitude is represented by the length of the arrow It s best to use a scale simi lar to those used for maps For example a displacement of 5 km might be represented in a diagram by a vector 1 cm long and a