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Atlanta University Center DigitalCommons@Robert W. Woodruff Library, Atlanta University Center ETD Collection for AUC Robert W. Woodruff Library 8-1-1958 Some applications of vectors to the study of solid geometry Ossie Malinda Smith Atlanta University Follow this and additional works at: http://digitalcommons.auctr.edu/dissertations Part of the Mathematics Commons Recommended Citation Smith, Ossie Malinda, "Some applications of vectors to the study of solid geometry" (1958). ETD Collection for AUC Robert W. Woodruff Library. Paper 2407. This Thesis is brought to you for free and open access by DigitalCommons@Robert W. Woodruff Library, Atlanta University Center. It has been accepted for inclusion in ETD Collection for AUC Robert W. Woodruff Library by an authorized administrator of DigitalCommons@Robert W. Woodruff Library, Atlanta University Center. For more information, please contact [email protected]. SOME APPLICATIONS OP VECTORS TO THE STUDY OP SOLID GEOMETRY A THESIS SUBMITTED TO THE FACULTY OP ATLANTA UNIVERSITY IN PARTIAL PULFILLMENT OP THE REQUIREMENTS FOR THE DEGREE OP MASTER OP SCIENCE BY OSSIE MALINDA SMITH DEPARTMENT OP MATHEMATICS ATLANTA, GEORGIA AUGUST 1958 if 1 -r ACKNOWLEDGMENTS * "" 36 The writer acknowledges her indebtedness to those instructors who made the writing of this thesis possible. Among those who should be mentioned are the late Dr. Gokhale, Mrs. G. C. Smith, Dr. V. Bohun, and Dr. L. V. Cross. I am grateful to each for whatever contribution was made. ii TABLE OF CONTENTS Page ACKNOWLEDGMENTS ±1 LIST OP FIGURES iv Chapter I. INTRODUCTION ! II. FUNDAMENTAL DEFINITIONS Definition of a Vector. * Vector Symbolism Negative and Null Vector.... Equality of Vectors Geometric Representation of a Vector... Reciprocal Vectors 3 3 3 3 4 4 4 III. ALGEBRAIC OPERATIONS Vector Addition, Subtraction of Vectors Scalar Multiplication Collinear and Coplanar Vectors Decomposition of Vectors 7 7 8 IV. VECTORIAL OPERATIONS Product of 5Bwo Vectors Triple Products Vector Product of Pour Vectors 9 9 10 11 V. GEOMETRIC APPLICATIONS Applications to the Sphere Applications to the Tetrahedron VI. CONCLUSIONS 5 5 6 12 12 13 17 VII. BIBLIOGRAPHY 18 iii LIST OP FIGURES Page 1. A Vector. 5 2. Negative Vectors 3 3. Reciprocal Vectors. 4 4. Vector Addition. 5 5. Demonstration of the Commutative Law... 5 6. Demonstration of the Associative Law 6 7. Subtraction of Vectors 6 8. Cross Product of Two Vectors 10 9. A Sphere with Center, C, and an Arbitrary Point, P, on the Circumference 11 10. Diagram Used to Prove That the Joins of Opposite Edges of a Tetrahedron Bisect Each Other. 14 .11. Diagram Used to Show That the Relation PA' - PB« - PC' - PD» - 1 Holds if P Is a TP Mr CU' EE Point within Tetrahedron ABCD and A», B', C, and D' Are the Meets of the Lines Joining the Vertices to P and the Opposite Paces iv 16 CHAPTER I INTRODUCTION We differentiate between two types of mathematical quan tities; those having magnitude only are called scalars; those having both magnitude and direction are called vectors. In this paper some of the elementary properties of the latter and their application to three dimensional space will be presented. Vectors arose as a result of the mathematicians1 attempts to represent complex numbers geometrically, A first step in this direction was the invention of the quaternion in 1843 by Sir William Hamilton, He made use of four quantities in rep resenting the quaternion: units, i, 3, and k. a + bi + e5 + dk. the units: one scalar and the three imaginary The quaternion, q is written in the form The following laws govern multiplication of i2-J2:k2: -1; U = -3i r. k; fr = -*$ : H ki r -ik - j. Quaternions are easily adapted to matrix algebra. matrix (a 4 bi c 4 di\ -c + di a - bi/ ) The may also be written in the following i <V If the coefficients of b, c, and d are called i, j, and k respectively, we obtain the form a+bi+oj+dk. The products 2. of the units in this expression behave ;just as those of the laternions of Hamilton. lUaternions f As an example, we see that ij r !Ehe use of quaternions was not too widespread, for the applied mathematician found them too general and complex, Josiah Willard Gibbs, an American, used some of the ideas introduced by Hamilton and rejected others in developing the field known as Vector Analysis. This paper is an attempt to show the properties of vectors, apply these properties to problems in three dimensional space, and draw some conclusions from this study. CHAPTER II FUNDAMENTAL DEFINITIONS Definition of a vector.- A vector is defined as a quan tity having both magnitude and direction. Associated with every vector ar© two points- the initial point or head and the terminal point or tail. Graphically, we show the vector PQ as the directed line segment from P to Q, ■p Figure 1 Vector symbolism.- The vector above may be indicated by the symbol P§ in which the magnitude is the scalar quantity PQ , and the arrow shows the direction to be from P to Q. In the proofs of theorems, however, this method becomes unwieldly; thus, hereafter a vector will be denoted by a single small letter with a bar to distinguish it from a scalar quantity. Negative and null vectors.- The set of vectors which have the same magnitude but opposite direction of any given vector is called the negative of the given vector. 6L Figure 2 3. Thus, in 4. figure 2, s« = -a*. In the case Mien the initial point and the terminal point coincide, then we have what is called the zero or null vector, The magnitude of this set of vectors equals zero* Equality of vectors.- Equal vectors are defined to "be those having the same magnitude and the same direction. This means that all equal vectors are parallel* Geometric representation of a vector,- We represent a vector on a Cartesion diagram "by plotting the coordinates of its initial point and those of its terminal point. If the coordinates of the initial point and terminal point of a are ( x , y , z ) and (x . y , z ) respectively, then -me differences xg - \$ y2 - yx» a*1* \ " \ are oai:Led the components of a. Two vectors are equal if and only if their corresponding components are equal. Reciprocal vectors,- If a' is parallel to a but a' • - , _____*___-__—_———•»—■— a, then a*1 and S are said to be reciprocal vectors. figure 3 CHAPTER III ALGEBRAIC OPERATIONS Vector addition,- Tha algebraic operations of addition, subtraction, and multiplication may be performed on vectors. We define the sum of a and b" in the following manner; the vectors, without changing their directions, place so that the terminal point of a coincides with the initial point of b~; the sum is the vector whose initial point is the head of a and whose terminal point is the tail of F. 1 Hgure 4 By using the definition we can show that the commutative law for vector addition holds, i.e., a + b" s b" + a. of a and b" is c\ The sum By completing the parallelogram in figure 5, we get b" - a - c also; therefore, jfche law holds. 6. Ihe associative law for vector addition states that for any vectors, a, F, and c", a f (F 4 c) s (a 4. F) + "5. 6, (F ♦ c) = cT and a ♦ 3 - 5. c s e, Thus, In figure Likewise (a ♦ F) s T and T 4 the associative law holds. t KLgure 6. The sum of several vectors is obtained by using the definition repeatedly. This sum is the vector whose head is the initial point of the first addend and whose tail is the terminal point of the last addend. Subtraction of vectors.- Subtraction of vectors may be considered as the special case in which we add a vector to the negative of another vector. Then a - b = a + (-b). Gibbs £l;12j gives a neat geometric interpretation of the subtraction of two vectors. are parallelograms. and a f (-F) » <T. In figure 7» OAB'D and OABC By the definition of addition, a+b s c However, <T equals and is parallel to e; therefore "a - b~« e". /SL 7. The rule states that the difference between two vectors drawn from the same origin is a vector whose initial point is the tail of the vector representing the subtrahend and whose terminal point is the tail of the vector representing the minuend. Scalar multiplication.- The product of a scalar quantity, s, by a is a vector which has been increased s-times in magni tude. If s is positive, the same as that of a; opposite that of a. s s 0. the direction of the new vector is if s is negative, the direction is The product sa yields the zero vector if When a vector is multiplied by a scalar, each of its components is multiplied by that scalar. Both the right and left distributive laws hold for the multiplication of a vector by a scalar,i.e., a(s + s ) 12 (s 4 s )a s - s a + s a. 12 Collinear and coplanar vectors.- If there exist scalar quantities, s and t(not both zero), such that sa + tF - 0, then a and F are called collinear vectors. In order for three vectors to lie in the same plane they must be linearly dependent. This says scalar quantities, r, s, and t (not all zero), must exist such that ra + sE + tc =0. Pour vectors are always linearly dependent. 8. Decomposition of vectors.- Coffin jT2;21 J shows that any vector may be decomposed into any number of component vectors which may or may not lie in the same plane. A most convenient selection is three mutually perpendicular axes, x, y, and z, along which are placed the unit vectors I", J, and E". A vector, a, with components a , a . a 12 3 as a r -fa J 4 a £, a quite useful form. 12 3 , can then be expressed The vector is then said to be a linear combination of the unit vectors. CHAPTER IV VECTORIAL OPERATIONS Product of two vectors.- In multiplying one vector by another, one encounters two types of products, scalar and vector products. These products have no analogue in the algebra of real numbers. The scalar product, sometimes called the inner product, is indicated by a raised dot; the vector product, also known as the cross product is indicated by an x. We define the inner product of two vectors as the product of their lengths multiplied by the cosine of the angle between them. By using the definition we get the following rules for multiplication of the units: 3E * EJ * O; W - k¥ - 0. I =7 = E - lj iq sji s 0; If two vectors, a and F, are expres. sed as linear combinations of their components and the unit vectors, we get the scalar product in the following way: (a.F) - (a T ♦ a 7 +a Ic).(b t + b 7 + t) E). 1 2 3 1 2 3 combining terms on the right gives (a b may conclude Multiplying and * a b •» a b ). We that the scalar product of two vectors is the sum of the products of corresponding components. The cross product of a and F is the vector whose magni tude is ab sin 9 (where 9 is the angle between a and F) and whose direction is the perpendicular from the plane of a and F viewed counterclockwise(Figure 8). 9. 10. Multiplication of the units is given by the following rule: I * J -I5E - 'J, = £ -<£? 3PJ - -$T = E; /Note that I ^ = r X x T etc.) All of the laws governing multiplication of real numbers do not hold for the latter product, 29ais product is not com mutative, but the distributive and the scalar associative laws are valid. (ax*) Figure 8, iEriple products,- It is possible to multiply three vectors and obtain either the triple scalar or the triple vector pro duct* lor any three vectors, a", "E>, and c, expressions of the type a(Fxe") and (axF)c are known as triple scalar products of a, F, and c", Shis product can be written in terms of the components of the three vectors. pression, we get (a.r (e_r 4 ©pi * cJs:)J. By expanding the first ex + a«7 f aj£)-j Cb,r 4 bgj 4 b-lc) x Simplifying the expression in the bracket we obtain (a I 4 a]u fc)./(bc -be )T + 1 2 3L23 32 (bo -be )] Mb c - b c )Tsl 31 13 12 21 Ihis last form reduces to a(be-bc)4a(bc-be)*a(bc-be), 123 3 2231 13 312 21 Asa deter- minant the scalar product takes the following forms 11. b a a 3 b 3 c 3 for any three vectors a, F, and c, an expression of the form ax (F x c) is called a triple vector product. This product could also be expanded in terms of its components to give the form (a b + a b (a c 4& c 4a © )(kfci +^0 + b^k) - 1 1. 2 2. 3 3- l + a b )(c i * o j4 c,k)» 5 % Mor® ooncisely, this is written in the form(a»gb -(a*tje. Vector product of four vectors.- She importance of the triple scalar product lies in the fact that all higher pro ducts can be reduced to the triple product. vector product (axb)x(cxd). Consider the By making use of a reduction formula we may write the following two expressions: c(abd)- d(abo) and b(acd) - a(bcd). two When combined, these expressions give a(bcd) - b(acd)+c(abd) - d(abc) - 0, whioh is the condition for four vectors to be linearly de pendent/, 5;12 J . CHAPTER V GEOMETRIC APPLICATION A few problems involving well-known properties of the sphere and tetrahedron have been solved as a means of showing how vectors are applied to Solid Geometry. Problem I; Express the equation of a sphere in vectorial form.- A sphere is completely determined if its radius and "Hie position of its center are known. In the sphere, S, (ligure 9) suppose P is a point on the sphere and C is the center. Let the coordinates of P be (p:., p , p 1,, 2 C be (c , c , c ). 12 (p - c ) 22 (p 3 2 2 - c ). 3 We may then writeISFf- iaj= 7 (p 3 - (p 3 ) and those of ll , -c ) 3 2-re & / , or/al- J ' (p 2 L - c ) i (p 11 1 -c ) 2 1 2 - c ) + 22 This is the equation of the sphere in terms of vectors. ilgure 9 12. - 13. Problem II: Show that the joins of the midpoints of opposite edges of a tetrahedron bisect each other.- Let A, B, 0, and D be the vertices of a tetrahedron, and let a, b", c", and <I be the respective position-vectors of the vertices with respect to an arbitrary origin, 0, Let X, m, n, u, v, and w represent the position-vectors of the midpoints of the edges with respect to 0 (Figure 10). Adding vectors a and b" gives the sum AB since the diagonals of a parallelogram are equal. However, AB ■- 2n; therefore n-_a__£_F. 2 obtain these additional equations: In a similar way, we . I - b" + c, m - c" + at u - a » <I , v - b" 4 cT , and w - c" f cC. !Qie limes which we want to bisect each other are LU, HW, and MY. The midpoints of these lines are the points of intersection of the diagonals of parallelograms OLDX , ONWX , and OMVX respectively, where 12 3 X , X , and X represent the unknown vertices of the completed 1 2 3 parallelograms. It is now necessary to show that these mid points are coincident. are Ihe position-vectors of these points Iu , 1+v, and n ■» w, each of which is equal to the expression •a»T)fc»dT Therefore, the lines bisect each other. 14. V T> \ O ELgure 10. 15. Problem III. Show that the expression PA» AA* Jjj7 s 1 PB« PC« + —— + BB» 00' holds if P is a point within tetrahedron ABCD and A1, B1, C, and D' are the points where the lines joining P to each vertex of the tetrahedron intersect the opposite face,The four vectors involved may be reduced to three by choosing an arbitrary vertex, say A, as the origin, and then writing b" s AB, c s AC, and 3 = AD (Figure 11). AP5p can be written as a linear combination of b", c, and <T; then p - rF 4sc 4 tcf, where r, s, and t are not all zero. Since AA» - a1 is a scalar multiple of p, a' = k(rb" * so 4> tcT); also a1 - F 4 BA>, Since BA* is coplanar with (c - b) and (d - b), it may be expressed in terms of them. Then a1 = m(c - b) 4 n(d - "b) 4 b. equating the coefficients of b, c, and d from the two expres sions of a', we obtain k - 1 - m - n r which when combined give k <- » k - m ~ s • 1 •» that PA1 s. AA1 - AP. PA1 _ k - 1 -l-(r + s + t). AA« " k we find that PB_ _ r BB» - (k - lJrfT + so 4 t?). Using the same method PCJ. . s, ^d PD_ . t. 00* - % Substituting the values of AA1 and AP and simplifying, we find that PA1 Then and k - n Erom the figure we "FTTTT * see By DD* four values gives the required results. Addition of these 16 fi CHAPTER VI CONCLUSIONS Since quantities such as force, velocity, acceleration, etc* have direction and magnitude associated with them, the applied mathematician will find vectors to be a most desir able aid. The pure mathematician is able to revel in the fact that the conversion of theorems from conventional form to vectorial form is possible; but this conversion, in many cases,is not a simplification. Both the usefulness and beauty of vectors, when considered together, serve to make them an integral part of the field of mathematics. 17 BIBLIOGRAPHY 1 Josiah Willard aibbs, Vector Analysis, lew Haven, Conn., Yale University Press, 1913 2 Joseph George Coffin, Vector Analysis, lew York, John 3 Albert Potter Wills, Vector and Censor Analysis, Hew Wiley & Sons, 19051 York, Prentice-Han, inc., 19)1 4 Homer E. Newell, Vector Analysis, Hew York, McGraw-Hill 5 Dr. Lonnie Cross, Book Company, liiu., 19 55.— " Hotes for Introduction to Higher Geometry", 1957. 18 (Mimeographed)