1.1 types of vectors (1)

1.1 types of vectors (1)

• 1.
  1.1 - 1 1.Vectors and Vector Operations 1.1 Types of vectors Three types of vectors that we will be concerned with are the following. 1. numeric vectors – lists of numbers 2. geometric / physical vectors – things with magnitude & direction, e.g. directed line segments, displacements and forces 3. Functions, e.g y = x2 . The connection between these different types of vectors is that for each type one can add and subtract objects of that type and multiply them by numbers. The operations of addition, subtraction and multiplication by numbers have the same algebraic properties in each case. Let’s take a look at these three different types of vectors. 1.1.1. Numeric vectors In this context a vector is a list of numbers. For example, 4 -1 7 is a vector. Often the numbers in the list are related in some fashion; we shall see examples of this as we go along. The vector 4 -1 7 has three components. The first component is 4, the second component is -1 and the third component is 7. A vector can have any number of components. For example, 4 -1 7 0 -8 is a vector with five components. Usually the order of the numbers in the list is important. For example, 7 -1 4 is a different vector than 4 -1 7 . When working with vectors, a number is sometimes called a scalar.
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  1.1 - 2 Thecomponents of a vector can be variables or formulas that represent numbers. For example, we would regard x y z and x2 + 3 -7 cos(y) as vectors with the understanding that we get 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 denote the vector 4 -1 7 . We indicate this by writing x = 4 -1 7 . When we use a letter to denote a vector, then a subscript on the letter indicates the particular component of the vector. For example, x2 indicates the second component of x. For example, if x = 4 -1 7 then x2 = -1. When we write the components of a vector in a column as with 4 -1 7 , it is called a column 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 -1 7 T = (4, -1, 7) and (4, -1, 7)T = 4 -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 ethernet cards sound cards resistors 5 7 capacitors 2 3 transistors 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 = vector containing the number of resistors, capacitors and transistors in an ethernet card
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  1.1 - 3 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 1.1.2. Geometric / physical vectors If you ask a physicist what a vector is he will tell you A vector is something with magnitude and direction A good example of a vector of this type is a directed line segment. To construct a 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 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. Often we regard two directed line segments as representing the same vector if they have the same length and direction. Suppose the directed line 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 same vector. This can be confusing since the actual directed line segments PQ and RS are different. P Q P Q R S
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  1.1 - 4 Inorder 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. Suppose the coordinates of P are x1 y1 . We shall indicate this by writing P = x1 y1 . Suppose also that Q = x2 y2 . Then to the directed line segment vector PQ corresponds the numeric vector x2 - x1 y2 - y1 . We shall indicate this by writing PQ = x2 - x1 y2 - y1 . 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 Q, so that PQ = x y . Example 1.1.2. Suppose P = 2 1 and Q = 5 2 . Then PQ = x2 - x1 y2 - y1 = 5 - 2 2 - 1 = 3 1 . 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 and PQ = x2 - x1 y2 - y1 and RS = x4 - x3 y4 - y3 . Then the triangles PQN and RST are congruent. So x2 - x1 = x4 – x3 and y2 - y1 = y4 – y3. So PQ and RS are both assigned the same numeric vector. So directed line segments that represent the same P = ( )x1 y1 x y x1 x = x2 – x1 x2 y1 y2 PQ = ( )x2 - x1 y2 - y1 = ( )x y Q = ( )x2 y2 y = y2 – y1 P = ( )2 1 x y 1 x = 3 PQ = ( )5 - 2 2 - 1 = ( )3 1 Q = ( )5 2 y = 1 2 1 2 3 3 4 4 5 x y x1 x2 y1 y2 P = ( )x1 y1 PQ = ( )x2 - x1 y2 - y1 Q = ( )x2 y2 R = ( )x3 y3 RS = ( )x4 - x3 y4 - y3 S = ( )x4 y4
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  1.1 - 5 geometricvector correspond to the same numeric vector. Forces. Forces are another 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 are P = x1 y1 = f cos( ) f sin( ) . So we can associate to F the numeric vector f cos( ) f sin( ) . Often we just write F = f cos( ) f sin( ) . For example, if f = 40 and = 30 , then F = 40 cos(30 ) 40 sin(30 ) = 40 3/2 40(1/2) = 20 3 20 = 20 3 20 = 34.64 20 . 1.1.3. Functions In more advanced work functions are often regarded as vectors. This includes functions y = f(x) of a single variable, e.g. y = f(x) = x2 . It also includes functions of several variables w = f(x, y, z) such as w = f(x, y, z) = xey + sin(z) and other types of functions. In the next section we shall see what functions have in common with numeric vectors. P = ( )f cos f sin = ( )40 cos 30 40 sin 30 ( )34.7 20 = 30 x y 10 f cos 34.7 f = 10 20 10 20 30 40 f sin = 20