Physics Trinity Valley School Page 1 Lesson #3: Part 1 Vectors and Scalars: A scalar is a number that represents a quantity of something. A quantity that can be represented using a scalar and a unit is called a scalar quantity. We will be seeing plenty of scalar quantities. Examples of scalar quantities include: Temperature 373 K, Mass 8 kg, Time 18 s, Energy 122.0 J Distance 42 m, Area 136 m 2, Volume 9.62 m 3, Electric charge 1.6 x 10-19 C, Electric current 0.24 A Electric potential 1.5 V Density 1000 kg/m 3, Speed 12.4 m/s. Scalar quantities can be manipulated in equations using the well-known rules of ordinary arithmetic and algebra, with the proviso that the unit designation must always be included in the equation, as well. We may say that a scalar quantity has, or represents, a magnitude; some amount of something. When we add or subtract scalar quantities there are two rules that must be followed. a) the quantities must have the same units; b) when expressed in scientific notation the quantities must have the same exponent. When we multiply or divide two scalar quantities they do not need to have the same units, nor do they need to have the same exponents. A vector also represents the magnitude of something. Furthermore, vectors are used to represent quantities that have an inherent directionality. Scalars cannot do that. Typically, vectors must be used to describe quantities related to motion or force. The direction of the motion or the direction of the force is an important characteristic that cannot be separated from the magnitude of the motion or force. To understand motion and force, the magnitude and direction cannot easily be interpreted separately.
Physics Trinity Valley School Page 2 Vectors provide a compact means of expressing and manipulating both the magnitude and the direction. It is possible to compute in physics by treating the magnitude and the direction separately, and we will be doing plenty of that, but the concept of a vector provides an organizing principle for understanding events and effects that inherently exhibit both a magnitude and a direction. Examples of vector quantities (in two dimensions) include: Displacement 146.5 m 45, Velocity 13.95 m/s 133, Acceleration 3.24 m/s 2 38, Momentum 16.1 kg m/s 127, Torque 7.37 Nm 21, Force 200 N 268, Electric field 4.9 x 10 9 N/C 135, Magnetic field 1.20 T 90 Note that the angles indicating direction must be measured relative to a pre-agreed reference direction, otherwise known as a coordinate reference frame. These vectors all represent fundamental physical quantities and the concept of a vector is fundamental to developing your understanding and appreciation of these quantities and the underlying phenomena they measure. Because vectors are more than just numbers, it is not surprising that the rules for manipulating them differ from ordinary arithmetic. Working with vectors is a little bit harder than working with scalars. On the other hand, vectors provide a framework for understanding concepts and phenomena that scalars alone cannot easily describe. For doing calculations we will frequently separate the magnitude and directional components. However, you should not forget that both are integral to a complete description of the system under study and that both must be reassembled after the calculation in order to complete the solution. We re going to use vectors repeatedly throughout this course, but as this is only your first incursion we will be taking just a first taste of what vectors are like. It is fair to say that almost all the most interesting physics studied in the last 400 years revolves around investigating the vector properties of the world. Your efforts to understand vectors will be handsomely repaid.
Physics Trinity Valley School Page 3 Representations of Vectors: There are three ways to represent vectors. Remember that the concept of vector is an organizing principle for understanding events and processes in the physical world. Do not confuse the vectors we re discussing with the physical processes themselves. Vectors enhance our understanding, but they are not in and of themselves the primary focus of our attention. Understanding the physical world is our goal. Nevertheless, you will be seeing plenty of vectors in this course and you need to learn how to handle them properly. Before we start using vectors to represent physical quantities, you first need to understand a few things about vectors in the abstract. Vectors are mathematical constructs. They do not exist in the real world. We only use them to represent things that go on in the real world. A car moving at 60 mph is not a vector. Its velocity is also not a vector. Nevertheless, we will use a vector on many occasions to do arithmetic with velocities because vectors mimic the natural behavior of velocities. For example, if we crash two vehicles together we can find out how the combination of the two moves after the collision. We accomplish this by finding the sum of the vehicles momentum vectors. To save us the trouble of crashing two vehicles together, we represent each vehicle with the mathematical equivalents for such properties as mass and velocity. To accomplish that, we use a scalar quantity to represent the mass of each vehicle, and a vector quantity to represent the velocity of each vehicle. We can manipulate these representations instead of manipulating the vehicles to predict what will happen. The value of vector quantities is that long experience shows that vectors accurately predict certain physical quantities, like velocity, very well. To understand events in the physical world it is clear that we need to understand how to manipulate vectors in the mathematical world. So let s begin.
Physics Trinity Valley School Page 4 We can picture a vector as an arrow. It has both a length, called the magnitude, and a direction. As our example, consider a typical velocity vector. The length of a velocity vector represents the speed of the vehicle, while its direction corresponds to the direction the vehicle is moving. Thus, there is a one-to-one correspondence between the motion of the vehicle and the vector representing its velocity. Here are some arrows. We ll imagine, for now, that each represents a velocity vector. The length is proportional to the speed and the direction indicates the direction of a moving object. A long arrow represents a rapidly moving vehicle while a short arrow represents a slower moving vehicle. The only inviolate properties of the vector are its magnitude and direction. Because vectors are purely mathematical, we are free to move them about in any way necessary in order to perform our manipulations; provided we don t change the direction or the length in the process of sliding them around. Vectors, representations of physical quantities, can be presented three different ways; as arrows, in polar form, in rectangular form 13.95 m/s 133, and 33.7 i + 16.3 j m/s.
Physics Trinity Valley School Page 5 Vector Addition - Visualizing the Process with Arrows I. The parallelogram method. Slide two vectors until they are touching tail-to-tail. The sum of the two vectors is obtained by drawing a new vector diagonally from the point where the vectors touch to the opposite corner of the parallelogram. II. Head-to-tail method. Move two vectors until they are touching head-to-tail. The order of addition is irrelevant and there are two different ways to accomplish this. Both produce the same answer. To get the vector sum, draw the vector from the tail of the first vector to the head of the second vector. As the diagram shows, the order of addition is not important. Either way you connect them, you will get the same answer; same magnitude and direction. III. The component method. Break both vectors into x and y components relative to a convenient coordinate reference frame. Then, add the x-components together and the y-components together. This is the most powerful method of vector addition. Add the x-components together to get the x-component of the answer. Add the y- components together to get the y-component of the answer. Note in the example that the y-component of vector B will be a negative number. The signs are important any time we use a coordinate reference frame. Keep track of the signs of the components.
Physics Trinity Valley School Page 6 Vector Names When we use arrows to represent vectors, they don t really need names. We can use letters simply to provide a reference for the text. However, if we are going to make this a systematic presentation we need a convention for naming vectors so we can discuss them even when a picture is not present. When typing them we will use a square, boldface letter, e.g., v, A, F, etc. To resemble a textbook presentation you can imagine a small arrow written above each name. The magnitude, or length without the directional information, of a vector, A for example, is then usually written as A, or sometimes simply as A or as the magnitude of A. Converting Among Vector Representations Arrows provide a great way to visualize processes involving vectors, but they are a terrible way to try doing vector arithmetic. Only a draftsman could love doing arithmetic that way. Still, most of us imagine the process by imagining these arrows in our heads as we do the addition. The work on paper is quite a bit different. The polar form of vectors is even worse for doing arithmetic. The very important use for this particular representation for vectors is that most data for vector quantities comes naturally out of our laboratory measurements in polar form. If we measure a speed and a direction of travel, then we can write the velocity vector in polar form without doing any additional calculations. Just write the speed, followed by an angle symbol, followed by the direction angle. The component form of representing vectors is the only convenient one for doing vector addition. Therefore, our first step in the process of doing arithmetic with vectors is learning how to convert the other two representations into component form. If you have a vector arrow, the first order of business is to convert it to the polar representation. Measure your angles from the positive x-axis or from the due East direction. The three vectors in the figure are represented as: A = 15 225 = 15 135 ; B = 15 10 WofN = 15 100 ; C = 10 25 SofE = 10 25
Physics Trinity Valley School Page 7 Once you have a vector in polar form, the next step is to convert it to component form. This requires a bit of trigonometry. The figure shows the general case for a vector in the first quadrant. The x-component is M x = M cos θ The y-component is M y = M sin θ If you need to convert back to polar form from component form, and you will, then M is given by M = (M x 2 + M y 2 ) ½ The angle θ is given by θ = tan 1 (M y / M x ) (this step can be tricky, especially in quadrants II and III.) Once you find the components of the vector, you must write it in the proper form, M = M x i + M y j The new symbols in this component form of the vector, i and j, are known as unit vectors. The i points along the positive x-axis and the j points along the positive y-axis. These unit vectors are known as i-hat and j-hat. Vector Addition Doing the Arithmetic Once you get the vectors into component form, doing the arithmetic is easy. Simply add the i-components together and the j-components together. These sums will give you the i and j components of the answer. Any number of vectors can be added together this way. R = (33.7 i + 16.3 j) + (15.2 i 6.1 j) = (48.9 i + 10.2 j) = 49.95 11.8
Physics Trinity Valley School Page 8 Properties of Vectors Here is a formal mathematical presentation of the arithmetic properties of vectors. Vector addition is associative, therefore, A + (B + C) = (A + B) + C Vector addition is commutative, therefore, A + B = B + A [Review the methods above to convince yourself that this is true] Vectors can be multiplied by scalars, which can increase or decrease the magnitude of a vector but cannot change its direction. When you multiply a vector by a scalar you get a longer or shorter vector. Multiplication by a scalar is distributive, thus m(a + B) = ma + mb (Remember that ma and mb are new vectors. They differ in length from A and B, though they share the same directions as A and B.) (m + n) A = ma + na (m + n)(a + B) = (m + n)a + (m + n)b = ma + na + mb + nb = m(a + B) + n(a + B) Vector Subtraction Vector subtraction is just another form of vector addition. When we write A B, what we really mean is add A to the negative of B (ie B pointing-in-the-opposite-direction) Or put another way, add A to the opposite of B (ie B). So, all you need to do is turn B in the opposite direction (by adding or subtracting 180 ) and add it to A in any of the usual ways.
Physics Trinity Valley School Page 9 Here are some sample vectors and their negatives. Change the signs if the vector is in component form. If A = (+15.2 i + 6.1 j) then A = ( 15.2 i 6.1 j) If B = ( 15.2 i + 6.1 j), then B = (+15.2 i 6.1 j) Change the angle by 180 if the vector is in polar form. If C = 50 12, then C = 50 192 or C = 50 168 If D = 25 120, then D = 25 300 or D = 25 60 Point the arrow in the opposite direction if a vector is represented by an arrow.