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2.1 Vectors We will be concerned with two different types of vectors. 1. 2. Geometric / physical vectors – things with magnitude & direction, e.g. directed line segments, displacements and forces Numeric vectors – lists of numbers The two types of vectors are connected when we introduce a coordinate system and the geometric / physical vectors can be identified with ordered pairs or triples of numbers. Another connection between these two types of vectors is that one can add and subtract vectors of each type and multiply them by numbers. Let’s take a look at these two types of vectors. Geometric / physical vectors A vector of this type is something with magnitude and direction. A good example of a vector of this type is a directed line segment. To construct a Q directed line segment we pick two points P and Q and draw the line segment from P to Q. We put an arrow at the end of the line segment at Q. We represent the directed line segment from P to Q P by PQ or even just by PQ. Note that the directed line segment from P to Q is a different directed line segment from the directed line segment from Q to P, i.e. PQ QP . In physics one uses directed line segments to describe displacements. Suppose we have a moving object and we are studying its motion. We fix two times, a starting time s and an ending time t. If the object is at point P at time s and at point Q at time t, then its net motion over the time interval from s to t can be described by the directed line segment PQ . This is called the object’s displacement over the time interval from s to t. Note that the actual path of the object during the time interval may be different from directed line segment. The directed line segment acts as a summary of the objects motion over the time interval. Often we regard two directed line segments as representing the same vector if they have the same length and direction. Suppose the directed line Q segments PQ and RS have the same length and direction. We might write PQ = RS if we are in a situation where directed line segments with the same length and direction represent the S P R 1.1 - 1 same vector. This can be confusing since the actual directed line segments PQ and RS are different. In order to make a connection between directed line segment vectors and numeric vectors, we draw a coordinate system. If the directed line segments we are working with are all in the same plane, then we draw a two dimensional xy-coordinate system for that plane. If they don’t lie in the same plane, we draw a three dimensional xyz-coordinate system for space. For simplicity, let’s suppose for the moment that they all lie in the same plane. x Suppose the coordinates of P are y11 . y x We shall indicate this by writing P = y11 . x Suppose also that Q = y22 . Then to the directed line segment vector PQ x -x corresponds the numeric vector y22 - y11 . We shall indicate this by writing x -x PQ = y22 - y11 . Note that x = x2 - x1 is the change in the x coordinate as we move from P to Q and y = y2 - y1 is the change in the y coordinate as we move from P to x Q, so that PQ = y . 2 Example 1.1.2. Suppose P = 1 and 5 x2 - x1 Q = 2 . Then PQ = y - y = 2 PQ = 2 1 2 1 Q= y2 (xy ) 2 2 y = y2 – y1 P= y1 x1 y1 () x = x2 – x1 x x1 x2 y 4 3 PQ = 1 5 - 2 = 3 . 2-1 1 (xy -- xy ) = (xy) ( 52 -- 21) = (13) Q= 2 (52) y = 1 P= 1 2 1 () x = 3 x Suppose PQ and RS are directed line segments and PQ = RS , i.e. PQ and RS have the same length and direction. Suppose we draw a coordinate system x2 - x1 and PQ = y - y and 2 1 x x 4 3 RS = y - y . Then the triangles 4 3 PQN and RST are congruent. So x2 - x1 = x4 – x3 and y2 - y1 = y4 – y3. So 1 2 3 4 5 y RS = R= 4 3 4 3 x4 S= y 4 () x3 y3 () PQ = y2 y1 (xy -- xy ) P= (xy -- xy ) 2 1 2 1 Q= (xy ) 2 2 (xy ) 1 1 x 1.1 - 2 x1 x2 PQ and RS are both assigned the same numeric vector. So directed line segments that represent the same geometric vector correspond to the same numeric vector. More generally, other physical quantities that have magnitude and direction are often regarded as vectors. We often represent these quantities by a directed line segment with the same direction and length equal to the magnitude of the physical quantity. Forces are a good example of a type of physical quantity that is a vector. If you push or pull on an object with a certain force, then the force F has a certain magnitude, f, and direction, . For example, suppose we pull on an object with a force of magnitude f = 40 lbs and we pull making an angle = 30 with the horizontal. Then we can represent this force by a directed line segment with length equal to 30 and making an angle = 30 with the positive x-axis. If we choose the starting point of the directed line segment equal to the origin, O, and the ending point equal to P, then the coordinates of P f cos() x1 f cos() . are P = y = . So we can associate to F the numeric vector f sin() f sin() 1 f cos() . For example, if f = 40 and = 30, then Often we just write F = f sin() 40 cos(30) 40 3/2 20 3 20 3 34.64 = = = = F= . 40 sin(30) 40(1/2) 20 20 20 Numeric vectors In this context a vector is a list of numbers. 4 For example, -1 is a vector. Often the numbers in the list are related in some fashion; 7 we shall see examples of this as we go along. 4 The vector -1 has three components. The first component is 4, the second component is 7 -1 and the third component is 7. A vector can have any number of components. For -1 example, 7 is a vector with five components. 0 -8 4 Usually the order of the numbers in the 7 4 list is important. For example, -1 is a different vector than -1. 4 7 When working with vectors, a number is sometimes called a scalar. 1.1 - 3 The components of a vector can be variables or formulas that represent numbers. For 2 x x + 3 y example, we would regard and -7 as vectors with the understanding that we get z cos(y) an actual list of numbers when we give numerical values to x, y and z. We often use a letter to denote a certain vector. For example, we might use the letter x to 4 4 -1 denote the vector . We indicate this by writing x = -1. When we use a letter to 7 7 denote a vector, then a subscript on the letter indicates the particular component of the 4 vector. For example, x2 indicates the second component of x. For example, if x = -1 7 then x2 = -1. 4 When we write the components of a vector in a column as with -1, it is called a column 7 vector. If we write the components of a vector in a row as with (4, -1, 7), it is called a row vector. Often it doesn’t matter if we write a vector as a column vector or a row vector, but as we go along we shall see some formulas where it makes a difference. If we are in a situation where we are distinguishing between column vectors and row vectors and we want to indicate that we are converting a certain column vector to a row vector or vice-versa, then we do this with the transpose operation. If x is a certain vector then xT = = transpose of x the same list of numbers as x, but written in row form if x is a column vector or written in column form if x is a row vector. For example, 4 T -1 = (4, -1, 7) 7 4 and (4, -1, 7)T = -1 7 Example 1.1.1. An electronics company makes two types of circuit boards for computers, namely ethernet cards and sound cards. Each of these boards requires a certain number of resistors, capacitors and transistors as follows resistors capacitors transistors ethernet cards 5 2 3 sound cards 7 3 5 There are a number of different vectors that might be of interest in a situation such as this. Here are some examples. u = 5 2 3 1.1 - 4 = vector containing the number of resistors, capacitors and transistors in an ethernet card v = (5, 7) = vector containing the number of resistors in an ethernet card and in a sound card y = 20 30 = vector containing the number of ethernet and sound cards the company plans to make this coming week r = 310 130 210 = vector containing the number of resistors, capacitors and transistors the company will need in order to make 20 ethernet cards and 30 sound cards this coming week week p = (2, 3, 5) = vector containing the prices the company has to pay (in cents) for resistors, capacitors and transistors 1.1 - 5