1CHAPTER ONEVECTOR GEOMETRY1.1 INTRODUCTIONIn this chapter vectors are first introduced as geometric objects, namely as directed linesegments, or arrows. The operations of addition, subtraction, and multiplication by ascalar (real number) are defined for these directed line segments. Two and threedimensional Rectangular Cartesian coordinate systems are then introduced and used togive an algebraic representation for the directed line segments (or vectors). Two newoperations on vectors called the dot product and the cross product are introduced. Somefamiliar theorems from Euclidean geometry are proved using vector methods.1.2SCALARS AND VECTORSSome physical quantities such as length, area, volume and mass can be completelydescribed by a single real number. Because these quantities are describable by givingonly a magnitude, they are called scalars. [The word scalar means representable byposition on a line; having only magnitude.] On the other hand physical quantities such asdisplacement, velocity, force and acceleration require both a magnitude and a direction tocompletely describe them. Such quantities are called vectors.If you say that a car is traveling at 90 km/hr, you are using a scalar quantity, namely thenumber 90 with no direction attached, to describe the speed of the car. On the otherhand, if you say that the car is traveling due north at 90 km/hr, your description of thecar's velocity is a vector quantity since it includes both magnitude and direction.To distinguish between scalars and vectors we will denote scalars by lower case italictype such as a, b, c etc. and denote vectors by lower case boldface type such as u, v, wetc. In handwritten script, this way of distinguishing between vectors and scalars must bemodified. It is customary to leave scalars as regular hand written script and modify thesymbols used to represent vectors by either underlining, such as u or v, or by placing anrrarrow above the symbol, such as u or v .
21.2 Problems1. Determine whether a scalar quantity, a vector quantity or neither would beappropriate to describe each of the following situations.a. The outside temperature is 15º C.b. A truck is traveling at 60 km/hr.c. The water is flowing due north at 5 km/hr.d. The wind is blowing from the south.e. A vertically upwards force of 10 Newtons is applied to a rock.f. The rock has a mass of 5 kilograms.g. The box has a volume of .25 m3.h. A car is speeding eastward.i. The rock has a density of 5 gm/cm3.j. A bulldozer moves the rock eastward 15m.k. The wind is blowing at 20 km/hr from the south.l. A stone dropped into a pond is sinking at the rate of 30 cm/sec.1.3 GEOMETRICAL REPRESENTATION OF VECTORSBecause vectors are determined by both a magnitude and a direction, they are representedgeometrically in 2 or 3 dimensional space as directedline segments or arrows. The length of the arrowQcorresponds to the magnitude of the vector while thedirection of the arrow corresponds to the direction of thevP
3vector. The tail of the arrow is called the initial point of the vector while the tip of thearrow is called the terminal point of the vector. If the vector v has the point P as itsinitial point and the point Q as its terminal point we will write v PQ .Equal vectorsTwo vectors u and v, which have the same length and samedirection, are said to be equal vectors even though they havedifferent initial points and different terminal points. If u andv are equal vectors we write u v.Sum of two vectorsThe sum of two vectors u and v, written u v is the vectordetermined as follows. Place the vector v so that its initialpoint coincides with the terminal point of the vector u. Thevector u v is the vector whose initial point is the initial pointof u and whose terminal point is the terminal point of v.uvu vvuZero vectorThe zero vector, denoted 0, is the vector whose length is 0. Since a vector of length 0does not have any direction associated with it we shall agree that its direction is arbitrary;that is to say it can be assigned any direction we choose. The zero vector satisfies theproperty: v 0 0 v v for every vector v.Negative of a vectorIf u is a nonzero vector, we define the negative of u, denoted –u, to be the vector whosemagnitude (or length) is the same as the magnitude (or length) of the vector u, but whosedirection is opposite to that of u.u uuuurIf AB is used to denote the vector from point A to point B, then the vector from point Buuuruuuruuurto point A is denoted by BA , and BA AB .Difference of two vectors
4If u and v are any two vectors, we define the difference of u and v, denoted u – v, to bethe vector u (–v). To construct the vector u – v we can either(i) construct the sum of the vector u and the vector –v; or(ii) position u and v so that their initial points coincide; then the vector from the terminalpoint of v to the terminal point of u is the vector u – v.(i)(ii)vvu vuu vu vMultiplying a vector by a scalarIf v is a nonzero vector and c is a nonzero scalar, we define the product of c and v,denoted cv, to be the vector whose length is c times the length of v and whosedirection is the same as that of v if c 0 and opposite to that of v of c 0. We definecv 0 if c 0 or if v 0.ParallelvectorsThevectorsvand cv are( 1)v½vparallel to each other.Theirdirectionscoincide if c 0 and the directions are opposite to each other if c 0. If u and v are parallel vectors,then there exists a scalar c such that u cv. Conversely, if u cv and c 0, then u andv are parallel vectors.v2vExampleLet O, A and B be 3 points in the plane.LetOA a and let OB b. Find an expression for the vectorBBA in terms of the vectors a and b.bOaA
5SolutionBA BO OA OB OA OA OB a–b.Example Prove that the line joining the mid points of two sides of a triangle is parallelto and one half the length of the third side of the triangle.CSolutionLet ABC be given. Let M be the mid point of side AC andlet N be the mid point of side BC. ThenMN MC CN 12 AC 12 CB 12 (AC CB) 12 AB .This shows that MN is one half the length of AB and alsouuuurthat MN is parallel to AB [since the two vectors MN anduuur1A2 AB are equal, they have the same direction and hence areuuuuruuurparallel, so MN and AB will also be parallel].MNBExampleLet M be the mid point of the line segment PQ. Let O be a point not on the line PQ.Prove that OM 12 OP 12 OQ .SolutionOM OP PM OP 12 PQPMQ OP 12 (PO OQ) OP 12 PO 12 OQ OP 12 OP 12 OQ 12 OP 12 OQO1.3 Problems1. For each of the following diagrams, find an expression for the vector c in terms of thevectors a and b.
6a.b.bccc.cbaabaB2. Let OACB be the parallelogram shown. Leta OA and let b OB. Find expressions forthe diagonals OC and AB in terms of thevectors a and b.CbOaA3. Let ABC be a triangle. Let M be a point on AC such that the length of AM ½length of MC. Let N be a point on BC such that the length of BN ½ length of NC.Show that MN is parallel to AB and that the length of MN is 2 3 the length of AB.4. Let the point M divide the line segment AB in the ratio t:s with t s 1. Let O be apoint not on the line AB. Prove OM s OA t OB .5. Prove that the diagonals of a parallelogram bisect each other.6. Prove that the medians of a triangle are concurrent.1.4 COORDINATE SYSTEMSIn order to further our study of vectors it will be necessary to consider vectors asalgebraic entities by introducing a coordinate system for the vectors. A coordinatesystem is a frame of reference that is used as a standard for measuring distance anddirection. If we are working with vectors in two dimensional space we will use a two
7dimensional rectangular Cartesian coordinate system. If we are working with vectors inthree dimensional space, the coordinate system that we use is a three dimensionalrectangular Cartesian coordinate system. To understand these two and three dimensionalrectangular coordinate systems we first introduce a one dimensional coordinate systemalso known as a real number line.Let R denote the set of all real numbers. Let l be a given line. We can set up a one to one relationship between the real numbers R and the points on l as follows. Select apoint O, which will be called the origin, on the line l. To this point we associate thenumber 0. Select a unit of length and use it to mark off equidistantly placed points oneither side of O. The points on one side of O, called the positive side, are assigned thenumbers 1, 2, 3 etc. while the points on the other side of O, called the negative side areassigned the numbers –1, –2, –3 etc. A one to one correspondence now exists betweenall the real numbers R and the points on l. The resulting line is called a real number lineor more simply a number line and the number associated with any given point on theline is called its coordinate. We have just constructed a one dimensional coordinatesystem.l 3 2 10123OTwo dimensional rectangular Cartesian coordinate systemThe two dimensional Cartesian coordinate system has as its frame of reference twonumber lines that intersect at right angles. Theyhorizontal number line is called the x axis and they axisvertical number line is the y axis. The point ofP ( x, y )xintersection of the two axes is called the origin andis denoted by O.To each point P in two dimensional space we associate an ordered pair ofyreal numbers (x, y) called the coordinates of thexpoint. The number x is called the x coordinate ofOthe point and the number y is the y coordinate of theorigin x axispoint. The x coordinate x is the horizontal distanceof the point P from the y axis while the y coordinate y is the vertical distance of the pointP from the x axis. The set of all ordered pairs of real numbers is denoted R2.
8Three dimensional rectangular Cartesian coordinate systemThe three dimensional Cartesian coordinate system has as its frame of reference threenumber lines that intersect at right angles at a point O called the origin. The number linesare called the x axis, the y axis and the z axis. To each point P in three dimensionalspace we associate an ordered triple of real numbers (x, y, z) called the coordinates ofthe point. The number x is the distance of the point P from the yz coordinate plane.The number y is the distance of the point P from the xz coordinate plane. The number zis the distance of the point P from the xy coordinate plane. The set of all ordered triplesof real numbers is denoted by R3. When the coordinate axes are labeled as shown in thezyyz coordplanexz coordplaneOxPyxy coordplanexzyOzxpoint P(x,y,z)following diagrams, the coordinate system is said to be a right handed Cartesiancoordinate system.Right handed Cartesian coordinate systemA right handed Cartesian coordinate system is one in whichthe coordinate axes are so labeled that if we curl the fingers onour right hand so as to point from the positive x axis towardsthe positive y axis, the thumb will point in the direction of thepositive z axis. [If the thumb is pointing in the directionopposite to the direction of the positive z axis, the coordinatesystem is a left handed coordinate system.]y1.4 Problems1. Draw a right handed three dimensional Cartesian coordinate system, and plot thefollowing points with the given coordinates.a. P (2, 1, 3)b. Q (3, 4, 5)c. R (2, 1, 2)d. S (0, 2, 1)
92. A cube has one vertex at the origin, and the diagonally opposite vertex is the pointwith coordinates (1, 1, 1). Find the coordinates of the other vertices of the cube.3. A rectangular parallelepiped (box) has one vertex at the origin and the diagonallyopposite vertex at the point (2, 3, 1). Find the coordinates of the other vertices.4. A pyramid has a square base located on the xy coordinate plane. Diagonally oppositevertices of the square base are located at the points with coordinates (0, 0, 0) and(2, 2, 0). The height of the pyramid is 2 units. Find the coordinates of the othervertices of the pyramid. [Assume that the top of the pyramid lies directly above thecentre of the square base.]5. A regular tetrahedron is a solid figure with 4 faces, eachof which is an equilateral triangle. If a regulartetrahedron has one face lying on the xy coordinate planewith vertices at (0, 0, 0) and (0, 1, 0), find thecoordinates of the other two vertices if all coordinates arenonnegativetetetrahedron1.5 DEFINING VECTORS ALGEBRAICALLYSince a vector is determined solely by its magnitude andydirection, any given vector may be relocated with respectto a given coordinate system so that its initial point is atvP(a,b)the origin O. Such a vector is said to be in standardvposition. When a given vector v is in standard positionthere exists a unique terminal point P such that v OP .xOThis one to one relationship between the vector v and theterminal point P enables us to give an algebraic definition for the vector v. If v is avector in two dimensional space and P(a, b) is the unique point P such that v OP , thenwe will identify the vector v with the ordered pair of real numbers (a, b) and write v (a,b). Similarly if v is a vector in three dimensional space and P(a, b, c) is the unique pointP such that v OP , then we will identify v with the ordered triple of real numbers(a, b, c) and write v (a, b, c). The two dimensional vector v (a, b) is said to havecomponents a and b and the three dimensional vector v (a, b, c) is said to havecomponents a, b and c.
10To avoid confusion, when dealing with the components of several vectors at the sametime it is customary to denote the components of a given vector by subscripted letters thatagree with the letter used to designate the vector. Thus we will write v (v1, v2) if v is avector in R2 and v (v1, v2, v3) if v is a vector in R3.Equal vectorsIf equal vectors u and v are located so that their initial points are at the origin, then theirterminal points will coincide, and hence the corresponding components of u and v mustbe equal to each other. Thus u v in R2 if and only if u1 v1 and u2 v2 while forvectors in R3, u v if and only if u1 v1, u2 v2 and u3 v3.Sum of two vectorsy2Let u (u1, u2) and v (v1, v2) be two vectors in R . Ifthe vectors are located so that their initial points are atu vthe origin, then their terminal points are the points v2with coordinates (u1, u2) and (v1, v2). If v is nowplaced so that its initial point is at (u1, u2), which is theterminal point of u, then the terminal point of v is the u2upoint with coordinates (u1 v1, u2 v2).u1Hence u v (u1 v1, u2 v2).A similar argument for the vectors u (u1, u2, u3) and v (v1, v2, v3) in R3 givesu v (u1 v1, u2 v2, u3 v3).vxv1ExampleLet u (1, 2, 3) and v (4, 1, 5). Then u v (1 4, 2 1, 3 5) (5, 3, 8).Multiplying a vector by a scalarIf u (u1, u2) is a vector in R2 that has its initialpoint at the origin, then the terminal point of u isthe point with coordinates (u1, u2). If c 0, then thevector cu has the same direction as u and is c timesas long as u so its terminal point is the point withcoordinates (cu1, cu2). A similar argument appliesif c 0, except in this case the direction is reversed.In either case we have cu (cu1, cu2).cu2cuu2uu1If instead u is a vector in R3, then a similar argument will show that cu (cu1, cu2, cu3).cu1
11ExampleIf u (3, 1, 2), then 5u (5 3, 5 1, 5 2) (15, 5, 10) .Difference of two vectorsThe vector u – v is defined to be equal to the vector sum u ( 1)v.If u (u1, u2) and v (v1, v2) are two vectors in R2, thenu – v (u1, u2) ( 1) (v1, v2) (u1, u2) ( v1, v2) (u1 v1, u2 v2).Similarly, in R3 we have u – v (u1 v1, u2 v2, u3 v3).ExampleIf u (4,5, 2) and v (2, 1,3) then u v ( 4 2, 5 ( 1), 2 3) (2, 6, 1).Vector representation of a directed line segmentLet v AB where A is the point with coordinates(a1, a2) and B is the point with coordinates (b1, b2).ThenAvv AB AO OBuuuruuuruuuruuur OA OB OB OA (b1, b2) – (a1, a2) (b1 – a1, b2 – a2).In R3, if A (a1, a2, a3) and B (b1, b2, b3) thenAB (b1 a1, b2 a2, b3 a3).BOExampleIf A (1, 2, 3) and B (4, 6, 9), then AB (4 1, 6 2, 9 3) (3, 4, 6)Length of a vectorIf v (v1, v2) then the length of v is equal to thelength of the directed line segment from the origin(0, 0) to the point (v1, v2). We will use the symbol(v1, v2)vv2v1
12v to represent the length of the vector v. Using Pythagoras’ theorem for right triangleswe can calculate that length to be v1 2 v 2 2 and so we have the formula v v1 2 v 2 2 .A similar argument for a vector v (v1, v2, v3) in R3, using Pythagoras’ theorem twice,222gives v v1 v2 v3 .Theorem If c is a scalar and v is a vector in R2 or R3, then c v c v .Proof The following proof is for v in R2. The proof for v in R3 is similar.c v (cv1 , cv2 ) (cv1 ) 2 (cv2 ) 2 c 2 (v1 v2 2 ) c 2 v12 v2 2 cv.Unit vectorIf v 1 we say v is a unit vector. Because the length of a vector is a positive quantity,the length of the vector cv is c v .To find a unit vector in the direction of a given11vector v, multiply the vector v by the scalar. The resulting vector v v is a unitv1vector in the direction of v. A unit vector in the direction opposite to v is v v .ExampleIf v (2, 2, 1), then the length of v is v 2 2 2 2 12 4 4 1 9 3 and a11 2 2 1 v (2, 2,1) , , . A unit vector in theunit vector in the direction of v isv3 3 3 3 2 2 1 direction opposite to that of v is , , . 3 3 3 1.5 Problems
13Let u (2, 1, 3), v (3, 1, 2) and w (4, 1, 1).1. Find the following vectors.a. u vb. u vw2. Find the following lengths.a. u b. vc. 2wc. 2wd. 2u 3vd. u ve. u 2v 3we. u vf. 2u 3v f. v w3. Find components of the vector equal to the directed line segment PQ .a.c.P (1, 2, 3) Q (2, 4, 7)P ( 2, 5, 1) Q (4, 3, 2)b. P (3, 1, 4) Q (5, 7, 1)d. P (0, 3, 2) Q (2, 0, 5)4. Let v AB . If v and A are as given below, find the coordinates of B.a. v (3, 5, 4) A (1, 3, 2)b. v (2, 5, 4) A (1, 2, 2)5. Let v AB . If v and B are as given below, find the coordinates of A.a. v (3, 5, 4) B (2, 5, 6)b. v (2, 5, 4) B (4, 1, 7).6. Let v be the given vector. Find a unit vector in the direction of v and find a unitvector in the direction opposite to that of v.a. v (2, 2, 1)b. v (3, 0, 4) c. v ( 1, 2, 3)d. v ( 2, 3, 4).7. If v (3a, 4a, 5a) and v 10, find the value of a.1.6THE DOT PRODUCT (SCALAR PRODUCT)The dot product is a method for multiplying two vectors. Because the product of themultiplication is a scalar, the dot product is sometimes referred to as the scalar product.The dot product will be used to find an angle between two vectors and will haveapplications in finding distances between points and lines, points and planes, etc.If u (u1, u2) and v (v1, v2) are two vectors in R2, we define their dot product, denotedu v , as follows: u v u1v1 u2v2.
14If u (u1, u2, u3) and v (v1, v2, v3) are two vectors in R3, we define their dot product tobe u v u1v1 u2v2 u3v3.ExampleLet u (1, 2, 3) and v (4, 5, 6).Then u v (1)(4) (2)(5) (3)(6) 4 10 18 32.The following theorem relates the length of a vector to the dot product of the vector withitself.Theorem For any vector u in R2 or in R3, u u u .Proof The following proof is for R2. The proof for R3 is similar.2Let u (u1, u2). Then u u (u1, u2) (u1, u2) u12 u22 u .Taking square roots gives u u u .The next theorem lists some algebraic properties of the dot product.Theorem Let u, v and w be vectors in R2 or R3, and let c be a scalar. Then(a) u v v u(b) c (u v) (cu) v u (cv)(c) u (v w) u v u w(d) u 0 0.Proof (a) Let u (u1, u2) and v (v1, v2) be any two vectors in R2.Then u v u1v1 u2v2 v1u1 v2u2 v u. The proof for R3 is similarThe proofs for parts (b), (c) and (d) are similar straightforward computations.The following theo