QA Z6\ H3 UC-NRLF B ^ s^fi a7S A Geometrical Vector \ Algeb By T. p! '1 t"V ) 1 1 ! ! ! ^_^fejj^ A WE.- VA'. '>->'"■.■>-,'.■ , HI A* A Geometrical Vector Algebra By T. PROCTOR HALL, VANCOUVER, CANADA CONTENTS Pagre 1. Introduction 1 2. Notation 1 3. Cosine AB 2 4. Addition and Subtraction... 2 5. Collinear Vectors 3 6. Coplanar Vectors 3 7. Multiplication 3 8. Collinear Multiplier 4 9. Perpendicular Multiplier 4 10. The Product AB 4 11. Permutation of Factors 5 12. Operand Distributive 5 13. Operator not Distributive. . 6 14. Factors not Permutable 6 15. Powers of Operator 7 16. Laws of Multiplication 7 17. Perpendicular Factors 7 18. Division 8 19. A"B; General Formula 8 20. Quaternions 9 21. Vector Arcs 10 22. Sum of Circular and Straight Vectors 10 23. Sum of Circular Vectors... 11 24. Spherical Triangle 12 Page 25. Conic Vectors 13 26. Differentiation 14 27. Differential Coefficient of A"B 15 28. Curvature 16 29. Linear Loci 16 30. Surface Loci 16 31. Solid Loci 17 32. Common Regions 17 33. Projections 20 34. Plane Algebra 21 35. Four-Space Algebra 21 36. Multiplication in 4-space... 21 37. Perpendicular Vectors in 4-space 22 38. Coplanar Equations 23 39. Perpendicular to a Vector 23 40. Normal to a Plane 23 41. Normal to a 3-flat 24 42. W, V, F and N 25 43. The Product ABC 25 44. Quaternion Rotors 26 45. Intersecting Loci 26 43. Projections 27 47. Projections of a Regular Tessaract on a 3-flat.. . 28 301858 A §0omctriral Brctar Alarbm By T. PROCTOR HALL, M.A.. Ph. D., M.D. 1. The laws of operation of any algebra are ultimately based upon its definitions. If the definitions are geometrical the algebraic operations have geometric correspondences. The operations of addition and subtraction in common algebra, for example, correspond to the geometric addition and subtraction of straight lines, vectors, surfaces, etc. In this algebra new definitions of vector multiplication and division are adopted, in consequence of which all algebraic operations upon vectors (directed unlocated straight lines or steps), or rather upon vector symbols, correspond to geometric operations in space upon the vectors themselves ; and every algebraic vector expression con-esponds to some geometric configuration of the vectors themselves. In every vector demonstration or problem, therefore, the student may think in terms of either algebra or geometi-y or both ; and may at any time change from one realm of thot to the other with no break in the continuity. This algebra is developed first in terms of analytical geometry for three-fold space, and is then adapted to two-fold and to four-fold space. Complex numbers, spherical trigonometry, and quaternion rotations, appear as special cases. 2. NOTATION.— Taking three rectangular axes X, Y, Z, let x, y. z denote unit vectors (steps) outward from the centre 0, along the axes. Unit vectors in the opposite direction from are denoted by x, y, z. Vectors in general are herein denoted by black faced Gothic capitals, and the corresponding unit vectors by black faced italics. For purposes of designation and operation all vectors (unless otherwise indicated) are under- stood to start from O, the centre of coordinates. Then if A is any vector, a is its length, a is unit length of the same vector, a^ x, a,, y, a, z aie tie vector components of A along X, Y, Z, and flx, Qy, fl, are the lengths of these components. Then A = aa — a^ X -\- Oy y -\- a^ z by vector addition. d^ ^= a'i -\- d'y -\- di by solid geometry. The symbol A is used to indicate (1) the vector from O to the point whose rectangular coordinates are «^, Oj, o, ; (2) motion from O to the extremity of A ; (3) a rotor, defined in ^7. Z A, GSOMETRICAL VECTOR ALGEBRA The Ine or locus of A is expressed by an elongated A thus /A, and any part of this locus, from m to n, is written ,,,/a. Surface loci are ordinarily expressed by two I's and solid loci by three is. 3. To express the cosine of the angle between two vectors in terms of the coordinates of the vectors. Let c be the length of the line joining the extremities of the vectors A. B, fi-om O. By solid geometry — c2 = (a, - Z>J2 + (o^ - Z>y)' + (o. - by By plane trigonometry — c'' = a- -\- b' - 2ab cos A B. a^ b^ -\- Gy b^ -{- a,, b.,. Sab Fig. 1. Therefore, cos A B = a b ab where Sab =^ the sum of the a b products ^ ab cos A B. If S,.i, = O, A B. and conversely. Example 1. — Find the angle between the vectors x + 2y and 2x-y-|-c^. Here S ^ 0, and the vectors are perpendicular. Example 2. — What angles does the vector 2x - y -(-z (= A) make with the axes x, y, z ? 2 -1 1 a = ^ G; .: cos A X = ^ Q , cos Ay = j 6 - cos Az = ^6 • 4. ADDITION AND SUBTRACTION.— Addition is geometrically defined as the piocess of making the second vector step from the extremity of the first. ^ he sum is the new vector from O to the extremity of the second vector thus added. Algebraically addition is performed by resolving the vectors into their components and adding these. A + B = (a, X + Qy y -[- a, z) + (A, x + Z>, y + b,z) = (fi^^bj X + UJy-\-b,.) y + {a,r{-bj z. Subti'action is addition of the negative of a vectoi-. Hence, both geometi-ically and algebraically vector terms are commut- ative. A B = B + A. A GEOMETRICAL VECTOR ALGEBRA 3 5. COLLINEAR VECTORS.— Two vectors A. B, are in the same line when A EEE nB, Ox 67 If n is positive, A and B are in the same direction ; if negative, A and B are opposite. \i n — \, A. ^^ B. 6. COPLANAR VECTORS.— Three vectors, A. B. C. are in the same plane when another vector, K, can be found which is peipendicular to each of them. Then S^k = S^k = Sek = 0, by i^3. Eliminating k^, kj,, k^ we get the coplanar equation \a,byC^\ = 0. The determinant ' a^ h, c, is six times the volume of the tetrahedion whose corners are O A B C. When this volume is zero A, B. C. are coplanar. Example. Find the conditions under which A is jjeipendicular to C= X + z, and in the B C plane where B - 2x - y, 3. The condition of perpendicularity is S,., = 0, or a^-\- a^ = 0. The coplanar equation is 2 -, 3 1 1 flx Oy Oi Ther^ore A = o^ (r - y, 3- z). = 0. 7. MULTIPLICATION of the vector B by the vector A is written A B, and is defined geometrically as the combined oi>erations, (1) Extension of B until its length is ab, (2) Simultaneous rotation of B thru 90^ about A as an axis, in a direction which is right handed or clockwise when facing in the positive direction of A. Each vector multiplier is a tensor-rotor. The rotor power of all vectors is the same and needs no separate expression at this stage. The product AB is that vector from O whose extremity is the final position of the point B after extension and rotation. The locus of AB is the curve traced by the point B during the operation. ABC means the operation of A on the product BC, or ABC m: A. BC. Also A" B ^ A. AB. etc. A GEOMETRICAL VECTOR ALGEBRA It next becomes necessary to find the laws of algebraic multiplication that correspond to the geometric changes here defined. 8. Multiplication by a collinear vector makes no change except in length or sign, XX = X XX = X XX = X X X = X Aa = A, etc. 9. Unit perpendicular vectors give the following results which are geometrically evident. xy — z ,xy = z y xz = y xz = y xy— z xy = z xz = y xz = y and similarly for y and z as operators. X Here the laws of signs are the same as in common algebra, so long as the factors are in alphabetical circular older; Fig. 2 xy = z ^ xy, xy ~ z ■= xy =— xy. But reversing the order of the factors changes the sign of the product ; xy — z yx = z. The second power of an unit perpendicular operator is equivalent to -1, x-y = xz ^ y x^y = X z = y. The fourth power leaves the operand unchanged, X V ^ x-y ^ y. When the vectors are not units the product of their tensors is pi-efixed to the vector product, ax. by^ab. xy — abz. 10. To find the algebraic product of any two vectors. Let the product be K = AB. Draw KD l O A, and OV equal and parallel to D K. Then the length OD is D = K cos A K =aZ> cos A B = Sab, bg >?8. As vectors 0K = 0D + DK, or K = S,,, a + V. Fig. 3 A GEOMETRICAL VECTOR ALGEBRA To determine V we have the equations of perpendicularity, Sav = o„ v^ -\- Oy Vy + a, v^ = o Sbv = ^x i'x + ^y i'y + ^2 i'z = O, and from the triangle O D K, i>l. -\- v\ -\- vl =^ V- ^ a-b- — S-. Solving we get y^ ^ Oj. Z>2 — Oj^by Vy ^= a^b^ — a„b^ v^ = a^by — Qyb^. Hence the "Vector Normal" to A. B, is V = QxOy Oz b.by b. X y z and its length is V = y a'ti'-^'' = ab sin AB. The product K is thus expressed in terms of the given vectors and their components, in the equation AB =r Sa + V. Example. Find the product AB when A=3y — Jf, B = 3*-y. Here S = -6, V = 8x, o = , 10, ... AB 8x -^--i?(3y-'2). 5 11. PERMUTATION OF FACTORS. It is geometrically and alge- braically evident that Sab ^^ Sba and that V.b = - Vb.- Hence B A -- Si,,. i> + V^, = Sab to - V.b which is not equal to A B. Changing the order of the factors changes the vector p»«oduct. Vectors are not permutable. 12. OPERAND DISTRIBUTIVE. To find the product A (B :- C) let B - C = D. so that x — Cx dy = by — Cy d^ = b^ ± Cj. 6 A GEOMETRICAL VECTOR ALGEBRA Then A (B + C) = AD -- S^d a + Vad = (Sab ± Sae) a + V,b ± V,e = (Sab a + V,b) ± (Sac a + V,,) -^ AB ± AC. The operand is therefore distributive. 13. OPERATOR NOT DISTRIBUTIVE. To find the product (A ± B)C, let A t B = K so that k, = a,± b, ky = Qy ±: by A^j = Oz ± K k^ = d" -\- b-" ±. 2 Sab, by §3. and Then (A B)C = KC = Sue #f + Vke Sa. ± Sb 1 a- + Z)^ zt 2 S. which is not equal to AC ± BC. Hence the operator is not in general distributive (A ± B) + (V. Vb.) 14. FACTORS MUST.NOT CHANGE ASSOCIATION. A B C = A (Sbc b + Vbc) = -^AB b AV. %-|^' A+^V3b+ U.A.c.la + SaeB-SabC. To expand A B. C, let K = A B, so that A B. C = K C = S,, If + Vk i Sab ^a a — -\- \ a^byC^ a b V. + Oj. b,, \ , \ a^ b^ Cx Cy X y Ox by c. which is not equal to ABC. Hence the association of a factor must not in general be altered. But if C = A these two products become identical, and therefore A. BA = AB. A. - A GEOMETRICAL VECTOR ALGEBRA 7 15. POWERS OF AN OPERATOR. AB = Sa + V A'B = A. AB = A (Sa + V) = SA + AV = 2 SA — a^B by expansion and multiplication. A^B - A (2SA- a-B) = 2 a^ S a - aM S a + V) = a^ (Sa — V), which is geometrically evident. A'B = a^ Sa - oMSa - a-B) = a* B, which is also geometrically evident. From these results it is easy to write the expansion of any value of A"B when /J is a positive integer. 16. LAWS OF MULTIPLICATION, summary. (1) Factors are not permutable (>;11) A B is not equal to B A. (2) The operand is distributive (?512) A (B it C) = AB - AC, but the operator is not (??13) (A B) C is not equal to AC ^ AC. (3) The association of a factor must not be changed (f;14). A. B C is not equal to A B. C but A. B A AB. A. (4) The fourth power of an operator is equivalent to the fourth power of its tensor (^15). (5) The common laws of signs are true for opera ' and product ; not for the operator. 17. PERPENDICULAR VECTORS. When A.B.C are perpen- dicular, S,i. = S^c = Sh, = o, (?$3), and AB = V, (jilS). Then the laws of J;16 become the following: (1) Permuting the factors, i.e., interchanging operator and operand, changes the sign of the product. B A = Vba = - Vab = - A B. (2) Both operator and operand are distributive. (A = B)C = AC rt BC A (B ± C) = AB = AC. 8 A GEOMETRICAL VECTOR ALGEBRA (3) The association of a factor must not be changed. A. B C is not equal to AB.C. but A. BA = AB. A. (4) The square of an operator is -1 times the square of its tensor. A'B = — a-B. (5) The common laws of signs hold true. If a, b, c be any three unit perpendicular vectors in the same circular order as x, y, z; then ab = c^ be =^ a, ca ^^ b, and these vectors may serve as units of the system, as well as x, y and z. 18. DIVISION is the inverse of multiplication, so that if AB = C C ^ = A-C=B. Geometrically, division is a negative turn of 90° about the divisor (identical in this respect with multiplication by the negative of the divisor) and reduction in length to that given by the quotient of the tensors. 19. GENERAL FORMULA for A" B where n is real. Let OA, OB, be the vectors A. B, and let O C be in line with their vector product, so that A" B = a" times OC. Let BCT be the circle of revolution of B about A; N its centre; N B, NO, its radii. Draw CD j NB; DE || BO. Let . CNB = (^ = ;7^ be the angle of rotation of B. Then NC = NB = -, where v is the length of V^,, n D C = ^ S//J 6, D E = B O ^g = ^ COS (y, OE=DBj;^ = ^ vers 0. cot A B = ^ vers B. - = ^ vers 0. A GEOMETRICAL VECTOR ALGEBRA 9 As vectors OC = OE + ED + DC S v = a vers d + B cos d + - sin 6 a a .: A" B = a". O C = a"' (Sa vers 6 + a B cos d + V sin $). This formula, being true for all real values of n, includes products, quotients, powers and roots of vector operators. Example. — Two rods, A and B, are joined at one end. A is one foot long, and the peip3ndicular distance of its free end from B is six inches. B is turned 60° about the axis of A, then A is turned 90° in the same direction about the new axis of B. Find the new position of A. Let the joined ends be at O. Let B = bx, and A a^ x -\- a^y. Since a — 1, and Oy = |, A = J (x i 3 + y)- The result of the first rotation is represented by C = A^B= Sa vers 60° + B cos 60° + V sin 60° == «' (7x + y , 3 - 2x , 3). The second rotation is c A = S c + V,, = 1 (9x , 3 — 3y - 2i), which gives the final position of the free end of A. 20. QUATERNIONS. When A J B, S = and V = AB. (f;17). Then A"B = a" (B cos ^ + a B sin ^) = a" (cos ^ a sin H) B. Now a- as a perpendicular operator is equivalent to -1; and by ex- pansion in series, exactly as with the complex {cos d -{- i sin H), it may be shown that the rotor of A" cos 6 ^a sin 6 = e**^ where is the angle and a the axis of rotation. , Hence for perpendicular vectors A"B = a"e"^ B. 10 A GEOMETRICAL VECTOR ALGEBRA The operator A" is a tensor-rotor-vector, or a directed quaternion, when applied to vectors perpendicular to A. It has the four funda- mental characters of a quaternion, namely. C, A" may be regarded as the ratio of (1) Since A" B C to B; (2) It is the product of a tensor and a directed rotor, a", e (3) It is the sum of a scalar or number and a directed unlocated line or vector, a" cos 8 -\- a" a si.-t 0; (4) It is a quadrinomial of the form k -{- I x -\- my -\- nz, where k is a pure number and the directive units x, y, z, have the relations x^ = y- = 2- = xyz = — I. 21. VECTOR ARCS. The rotor e*^ turns thru the angle a about the axis A any vector in the plane perpendicular to A. The index a is a vector angle whose axis is A and whose magnitude is a radians. The length of the subtended arc is a a. If this circular arc be taken as a vector, written a, it is understood that its angle is a, its axis A and its radius a. A vector arc may take any position in its own circle, and has therefore one more degree of freedom than its vector axis. Vector arcs need not be confined to arcs of circles, but whether the extension to other curves would be of any particular value remains to be seen. A rough classification gives the following: (1) Straight vectors, (2) Plane vectors, having single curvature. A. Conic, a. Circular, b. Elliptic, c. Parabolic, d. Hypeibolic, B. Spiral, etc. (3) Solid vectors, with double curvature. 22. SUM OF CIRCULAR AND STRAIGHT VECTORS. Let the plane of the arc a meet the plane of A, B, in the line CC; let C be so chosen C that ^ B C is not greater than 90°, i. e., so that n is positive ; and let a = c. A GEOMETRICAL VECTOR ALGEBRA 11 Let C = mJK + ajB. Then from the figure h- b' — nf d^ = c^ = = a-. ma cos AB= ^ = ab a' S .•. /J = — . m — - , u ' V (^3) •.C= y (S A + Q-^B). Let Then COD = a' ,a'+« y+a F - D = e" ^" C - c" C = (e" ' " - e" ) C = [ cos {a^ -\- a) -\- a sin (a' + a) — COS a^ — a sin «' ] C = 2 sin " , sin (a^-\- " ) A- a cos (a' + " ) , C. 2 ( ' ' 2 2 * To this B is readily added. If B is parallel to A, C is indeterminate and any radius of the a circle may be taken as C. In this case the sum is a point on a right helix or screw whose axis is A. Since the addition may begin at any point of the a circle, tha sum is a screw vector whose radius, pitch and direction aie fixed. 23. SUM OF TWO CIRC^ULAR VECTORS. Let a, B be two cir- cular vectors with a common centre O; and let C = V„,, be the intersec- tion of their planes. Let CBo = /i', C Ao = a'. Then a = A, B= B. Ao = (e'''+'* - £■''') a c, Bo= (e^'+^- e^')Ac. Any third circular vector whose position is deter- mined with reference to the intersection of its plane with the plane of a or B, may be similarly expressed and the sum readily found. In expanding these ex- pressions it is convenient to remember that Fig. 6 12 A GEOMETRICAL VECTOR ALGEBRA when C = V,^,, then AC = V„, = S^i, A — o= B and BC = Vi„ = A-' A — S„i, B. When S ^ a the sum is 2a = a. 2a which is a vector arc with angle a and radius 2a. The locus of the sum of two equal vector arcs beginning at the same point of intersection, when the planes are not identical, is an ellipse. Also I (a — a) is a straight line. 24. SPHERICAL TRIANGLE. Assume a sphere of unit radius, and upon it arcs of great circles. As an illustration of vector ti-eatment let it be required to find the relation between the sines of the angles of a spherical triangle. Let a, B, y, be three cir- cular vectors forming a spherical triangle ; a, b c, their vector axes; A', B'. C\ the vectors from O to the angular points; A, B, C, the angles of the spherical triangle. Fig. 7 Draw A' A2 j_ O C^ Then as vectors Pi} = On -\- n K^ or A' = C' + ^ = C' cos /■] + b',,,! sin ft. Similarly A' = B' — y = B' cos y — v^t,^ sin y. By inspection of the figure it is evident that in any spherical triangle a = v,i,i cos a = Sbici> sin a == i\,i^.\ cos A = — cos (- — A) = — Si,^. , sin A = tv.. I Z>., c, z I A' = Vu = O' Q'y y + O^ A GEOMETRICAL VECTOR ALGEBRA 13 Similar equations may be written for the corresponding elements of the triangle. From the last equation, equating coefficients of x, y, z. al = Similarly, b[ = - by c, I Cy a, I I b, c, I. fbc I gx by Then cos y = Saii,» Obc 0(~ b. ^bc i'.c and 1 1 - S^b« i/(l - SL - SL - SL + 2 S,, S,.. S„) S/>? C y...l. ^ab ^bc "cs The last expression is symmetrical in a, b, c. and therefore sin a *'^ /^ sin y sin A ~ sin B '~ sin C 25. CONIC VECTORS are expressible in teims of the radius vector from the focus to each extremity of the segment of the curve. m B Let A be the axis of a conic, O it.s focus, N its directrix, P the radius vector, a. b the coordinates of P with reference to A and B: and let p = e (a -\- m), where e = is the eccentricity. - p- (1 — c') + 2cmp — m'-. Fio. 8 Then a ~ cp — m b- = p^ — (f - .-. P ^ A + B ^ {cp - /;?) a + /b, \p- (1 — C-) + 2cmp The conic vector from Po to P is P — P 1 the curves are expanding and when a 1 diminishing. 30. EXAMPLES OF SURFACE LOCI. 77' „t„t(A r B) + C is a parallelogram whose adjacent sides A, B, start at the point C. Its diagonals are A :b B. Its area is a 6 sin A B = f . (SlO). A GEOMETRICAL VECTOR ALGEBRA 17 JJb"a B is a closed surface, spherical if a j b, with radius b. ,1 „l o" B is the conical surface traced by B as it is turned about A. ./:/ (A+ (1 — o) B) is the triangle OAB. 31. EXAMPLES OF SOLID LOCL /// (A + B + C) + D is any parallelepiped. Its diagonals are A + B -f C. A + B — C. A — B C, — A + B + C. Its volume is | o, b,. c, | . If P -- V,^ and Q =^ V^^, the dihedial angle, a, over the edge A is found from the equation S,„, = pg cos a. „/„/„/a"^'"A is a shell, spherical if A B. 7' \ 7' 7' ) <,/ a" ( „c ^'" e t C + R , is a hollow annulus if B C, R ' C. A R, A B. 32. THE REGION COMMON to two loci is found by equating the coafficients of x, y, z, in the expressions for the loci. If these equations are consistent, giving i"eal values for the variables, the limits thus found are inserted in either of the loci to give the required locus of intersection. Example \. — Find the region common to the straight line „l n X -\- hy, and the curve „/{a X + (2x + y) sin a j. Equating coefficients, n = a -\- 2 sin a i = sin a. Whence n = a -\- 1 =^ arcsin I -\- 1. Inserting these values, both loci become Jih y + X arcsin h) which is a row of discrete points parallel to :ir. 18 A GEOMETRICAL VECTOR ALGEBRA Example 2.— Find what part of the helix /" 7" mr nir ^ (X" y -\- 3/? X) ^ X (3 /JX +y cos — [- z sin — ) is within the figure nr. , . 7/7' '"^ '"'^. .,( d J (y"'c z + bx — y}^^ J J „l ,x (csin-Y- -{-b)—y+zccos -^ \- Equating coefficients of x, y, z, (1) 3/2 = c sin ~ + b (2) cos ^ = - 1, .-. sin — = 0, and n =2, 6, 10 (3) c COS — — sin — = 0, 2 2 If c = 0, 3n = A. It COS — - = U, s^n — - =^ ± 1, and since c is positive Sn ~ b -\- c. Inserting these values in the locus of the helix we get for the intersection a row of points ISnx — y, where n has the values 2, 6, 10 up to -^ ■ Example 3. — Find the intersection of the plane II {mx -\- nz) -]- Sx with the solid o/ „IJ X" («•« + by) ^^ot oU \ax -\- by cos -| hzsinO] ■ Equating coefficients of x, y, z, (1) /n + 3 = o, or /n = o — 3. (2) b cos B = 0, .-. A = 0, or sin 6 ^ ± I. (3) n = b sin 0, = or ± b. .-. n = ± b. Substituting in the locus of the plane we get for the inter- section the parallelogram o/„/(o bz). A GEOMETRICAL VECTOR ALGEBRA 19 Example 4. — Find the locus of the intersection of the cube JJjla^x^ Qy y — a^ z) with a plane which cuts its diagonal A = X + y 4- z perpendicularly. Let B = X — y be one vector in the perpendicular plane, and C = V.b =x + y -2z the other. The plane is 11(1 B mC+ nA) where n is an arbitrary constant expressing the fractional distance from O to the point where the diagonal is cut. Equate coefficients of x, y, z, in the two loci, a^ = I ' m -\- n Oy = — / -\- m -\- n Oj, = — 2 m ' n. Therefore / = — g— Ox + Oy — 2 o, m= g 3 /J = a^t 4 Oy + Oj. If /2 = the plane goes thru O. Since a^. a^, o^, are all positive and the sum zero, each of them is zero, and the locus of intersection is the point O. If n = 1 the point of intersection is A. If n = J, so that Ox ; Oy + Oj = 1, while each varies between and 1 subject to this condition, the locus is an equilateral triangle whose corners are found by giving to a^, a^, a,,, separately the maximum value, 1, in the expanded expres- sion for the plane If n = § the locus is a similar' triangle. If /? = I the locus a I'egular hexagon. 20 A GEOMETRICAL VECTOR ALGEBRA 33. PROJECTIONS. To express any vector K in terms of three non- coplanar vectors A. B, C, write /A H /n B + « C = K .•. i a^ + m b^ -\- nC:^ = k^ I Qy ^ m by + n Cy = ky I a^ -\- m b:^ + n c^ = k^ _ I k^ by c, I _ I k^ Cy a, | _ | Ar. o„ b, I Ox byC,\ ' ~ \ a^byC,\ ' " - \a,byC,\ • If we now write /? = 0, I Ai, -{- m B \& the projection of K, made parallel to C, upon the plane of A. B. If A and B only are given, and the projection is desired of K per- pendicularly upon A, B, take C = Vai, = \ a^ by z \ , and proceed as before. To project K in the direction of C upon a plane perpendicular to C, take any vector A, -L C, so that Sac = 0, as, A ^ CyX — c^ y and a second vector B, = Vacr B = Cx Cy C.^ Cy-Cx X y X Then express K in terms of A, B, C, as before. The most general form of a locus is /// K + M, which is projected in the same way. Example.— Project upon the YZ plane and parallel to D the helix I B ^^^ J {x"b y 4- an x) = J [b (y cos d -\- z sin 6) -\- on x } * Let B = /y + /7?2r ; rD. Equating the coefficients of x, y, z, r d^ ^^ a n I -{- rdy= b cos 6 m -\- r d^ ^^ b sin 0, .-. B = y j A cos 6 — a n -. ] -\- z \b sin ~ on-, \ the locus of which is the required projection. A GEOMETRICAL VECTOR ALGEBRA 21 34. PLA.NE ALGEBRA.. Every vector in the XY plane is of the form A = Ox ■* + «> Y = o^ X -\- Gy zx = (Ox + z Oy)x. Since x is a part of every vector expression of this form, it may be omitted. The remaining form, a^ -f- •* Oy. is a complex number. Since z'^ as a rotor is equivalent to -1, we may write this tensor-rotor in the common form o -|- ib (whare /- =-1), whose p.-operties are well known. Again, any vector in the X Y plane may be expressed as a z-product, thus, A = a z" X = a (cos 6 -\- z sin 6) x = a c'^ X. Omitting x as before we have left the other two forms of the com- plex number. Vector multiplication in the XY plane with any other rotor than z gives in general imaginary products, i.e., products lying outside of that plane. FOUR-SPACE ALGEBRA 35. In four-space there are, by definition, four mutually perpen- dicular axes, X, Y, Z, U. These are so selected that they multiply in cir- cular order, as in 3-space. Each vector is now fully defined by four compansnts. Vectors are added and subtracted as in 3-space. As in §3 it may be shown that Sab ^= Ox 6x + Oy Ay + Oz ^z + (7^, A„ = O Z> COS A 3, where S„b is, as before, the sum of the a b products. Evidently also when S^b = 0, A '_ B. 36. MULTIPLICATION in 4-space is defined as rotation about the plane* of the multiplying vectors, thru a right angle in the positive direc- tion. The planes of rotation are wholly + perpendicular to the axial plane. *Rotation is essentially plane motion. In a 2-flat the axis of rotation is a point. In a 3-flat the axis is a line. In a 4-flat the axis is a plane. tit is evident from §35, that in 4-space absolutely perpendicular planes exist. For A = a^ ■* + Oy y is any vector in the X Y plane, and B= b^z + b^u is any vector in the ZU plane. Since Sab= 0, A -i- B. That is to say, every vector in the XY plane is perpendicular to every vector in the Z U plane. 22 A GEOMETRICAL VECTOR ALGEBRA Multiplication of C by AB, is written ABC, and is defined as (1) Rotation of G thru 90" in the positive direction about the plane A B, and (2) Simultaneous extension to the length abc. By definition, xyz = u, yzu = x, zux = y, uxy = z. Re Timbering that the plane of rotation is perpendicular to the axial plane it becomes evident that xyu = z, yzx = u, zuy = x, uxz = y y xyz = u, yzu^^ x, zux ^ y, uxy = z lcyu = z, yzx = u, zuy = x, uxz=^y Fig. 9 Coplanar vectors are unchanged in position by 4-space multiplication, because the whole axial plane is unmoved, xyx = X xy (a^x -^r Qy y) = a^x -\- a^ y. 37. MULTIPLICATION BY PERPENDICULAR VECTORS. Let A -L B -L C, and let ABC = W- Then S,^ = Si,w = S,„ = and iv^ =^ iv'^ -\- w~ -\- wl + wl = ci^bi^ c^. Solving for w^, iVy, w^, w^, and collecting. Wab Ox Qy Oz Ou b.. by b. bn Cx Cy Ct. Cu X y z u Ox by c,,u \ . Also w'^^\ayb^c^\''-\-\ Wx Az c J " + Ox 6y c J - + I Ox by c. Saa S,, Sac Sba Sbb She Sea Seb Sec O" bi^ &. A GEOMETRICAL VECTOR ALGEBRA 23 33. C is coplanar with A. B, when C = m A - nB. Wiiting the four equations of coordinates, and eliminating m and n, we get the coplanar equations \ a„ by c^ \ = I Oy 6z c„ I = 0. 39. To find the perpendicular N- from the point B to the vector A. Q Let Q B be the positive direction of Me- Then ry = O Q = A cos A B = -" <^ = -aZ A. n- = b- — q^ =- Saa A B ^ a\ Sbb ^ 0^ = y2 a- These forms of N-, ri~, Q, q, are identical in space of four, three and two dimensions, and evidently for space of all dimensions. In a 2-flat I Ox Oy I f^ = \ h A H- a. Ox 0„ 40. To find the perpendicular N.j from C to the A B plane. Let Q C be the positive direction of Ns- Draw Q E _L A. Q F j_ B. join C F, C E. a ~ a : 830 = S., Then OE and F = -7- = -jr . b b A •'■ '5i,,| -= bbc Since A, B, Q3, are coplanar I Ox Ay q. ! = I o, Z>, (7J = (1) (2) (3) (4). 24 A GEOMETRICAL VECTOR ALGEBRA Solving for the coordinates of Q:i, and collecting terms, where Then Q3 (b- Sao — Sab S,,c) ^ + (a- Sbc — Sab S^c ) — 2 --^ Qi + Qy + qI ^ ql = (a^ SL + F- SL - 2 Sab Sa,. SbJ l-,, 0"- = a'-b" ~Sl^ N3 = C - Q3 Saa Sab Sac Sba Sbb Sbe ISaa ■ iSba Sab Sbb A B C Saa ^5ab Sao Saa Sba Sab Sbb Sba Sbb Sbi- Sea ^cb Sec w^ v' These forms are identical for 3-space, and apparently for all space above it. In 3-space also Ox 6„ c. 41. To find the normal N4, from D to the 3-flat of A, B.C. Let Q D be the positive direction of Nj- Join O Q. Drop perpendiculars from Q4 on A,B,C, and join each point of intersection with D. Then it is evident as in §40 that ^aq ^ '^ad '^) Sbq = Sbd ('i) Scq= S,d (3). Since A. B, C. Q4. are all in the same 3-flat and therefore all perpen- dicular to N4 San=Sb„=S,n-S,„=0. Eliminating the n's we get the cosolid equat'on I a^byC^q^ I =^0...(4;. N4V C A GEOMETRICAL VECTOR ALGEBRA 25 Solving for q^ etc. and collecting terms Q4 = { S,d [A ( SL - b' c^) + B (c^' Sab - S,, Sbc) - C ( A^ S,e - 8,^ S,e) ] + Sed [C (S!b - a- If) + A ib' S., - Sbc S^J + B (a^ Sbc - S,. S.J ] } ^ zi^^ Therefore N4= D- Q4 •^aa ^ab '^ac >5ad k^ba Sbb Obc Obd b,.a Ocb Sec Sjd A B C D Saa Sab Sac Sba Sbb k5bc Sea Scb s„ Saa ^bb S^e Jj I S,a S,,b S,e Ox by c^ d^ IV- 42. RELATION OF N4 TO THE RECTOR W. Since N4 and W are each perpendicular to the 3-flat of A, B, C. they differ only in their tensors, tIa I o, Z>y c, C = Qa vers 6* + C cos (^ + - sin d n vers if .: A B C = 0" b" [1 A ib^ S^ - S,h S,,,) + Bid' S,, - S,, S.J | — ^,— W + Cco6-f^+X^ ^..^^-j^ 44. If C is perpendicular to A and B. then " VV A B C =^ o" A" I ^ ^'"-^ ^ + T ^'" ^ ' ' W and ab C = — . .-. A B C = o" A" ( cr)s 6 + alb sin H ) C The rotor e" resembles the rotor e found in 3-space multiplica- tion. It is evident that similar rotors (quaternions) will be found in all higher space forms. 45. The intersections of loci are found as in S32. Example L — Find the intersection of the 3-flat /// (ax J^by + CZ) with the helix / I xy (jr -f- y + i) -f nx J :r:r [(x ■ y -\- z cos 6 -\- u sin t) + nx). Equating coefficients of x, y, z, u, a ^ n + 1 b ^ 1 c = cos = sin 0. .-. c = ± 1 and n = 0, rt 2, ± 4, etc. A GEOMETRICAL VECTOR ALGEBRA 27 The intersection is representing two rows of points parallel to X. Example 2.— Find the intersection of the plane Hiax i by) with the solid cylinder JJol [^ C + m (x + u) ) :^= lll[c^x - Cyy- (c, cos 6 — c, sin ti)z + (Cu cos B ~r c^si'n 6) m]. Equating coefficients of x, y, a — c^ -\- m b = Cy, and the plane locus becomes the rectangle 46. PROJECTIONS. To express any vector K in terms of any four vectors A, B, C, D, not in one 3-flat, write /A + mB + nC /D -= K. Then la^ -\- mb^ - nc\ -\- rd^ — k^ lOy -]- mby -{- nCy -\- rdy -- k, la^ -\- mbj. ■ nc^ -\- rd^ =: k^, la^-\- mb^ ■ nc^ -\- rd^ — ka Ox ky Cj (/„ I /= • / '7 . m = \ k^ by Cz rful 1 Ox by Cj <\ 1 Ox by k. rful I Ox by Cj, o'u I Ox by c^ ATu Ox Ay Tj <^U I ' ~ I Ox by Cj (/„ Writing either /, m, n or r equal to zero the remaining terms of K are the projection of K made parallel to the vanishing vector and upon the 3-fl it of the remaining vectors. To project K normally upon the 3-flat of A, B, C, write D = W;ibc, then make r = 0. The sum of any two terms of K is the projection of K ujjon their plane, made parallel to the plane of the other two vectors. Loci are projected in the same way. 28 A GEOMETRICAL VECTOR ALGEBRA 47. As an illustration of the method of the last section we may find the principal orthogonal projections* of the regular 8-cubed tessaract whose edge is unity, /;/:./;/k. upon the 3-flats about it. (1) Parallel to x, on the 3-flat of y, z, u, the projection is obtained by writing k^ = 0, giving the cube w\ (k,y-^Kz + k,u). (2) Parallel to x + y. Let x + y = D. To get three other rectangular vectors we may take A = z B = w C = Wabd = y — X. Then I — k,,, m — k^, n — -^—^ — ^ • Writing r = the projection becomes JJJ J] k.A + k^Bi^ "-"-^ C j . And a = 1, 6 = 1, c = i 2. To express this locus in geometrical terms we note first that since it contains three vectors, not coplanar, with independent variable coefficients, it is a 3-space solid ; and in the second place that the original axes, z, u, which are perpendicular to the line of projection, remain unchanged. The axes x, y, are each foreshortened in the ratio of i 2 : 1. Projecting x and y by the same plan as for K we get for the projections X' = iC making the total distance ; 2 along C. Consider next the variables in the locus, k^ and k^ are entirely independent, with limits from to 1, and [ [ (A", A ^ k^B) is a square in the A B plane. The solid is a right square prism whose extension along C is given by the last term ky - k, — 2 C of the locus, k^ and k^ vary independently fi'om *For a purely geometric investigation of these projections see the American Journal of Mathematics, Volume XV, No. 2, pages 179-189. A GEOMETRICAL VECTOR ALCEBRA 2Q (3) to 1. The lower limit of the term occurs when k^ = 0, k^ = 1, namely, -J C; and the upper limit is ^ C. Since c = ^ 2, the length of the prism is ^ 2 along C. Parallel to x + y - z, (= D). Take for the other rectangular axes A = u B = jf — y C = Wabd -= X - y — 2z. Then I = k^, m = k, - k. k. + ky 2 k, Put r = ; the projection is nii\K^ + k,-\-ky-2k. > "" " ' 2 where a = 1, 6 = ^ 2, c = ^ 6. The figure is again a 3-solid ; the axis u, perpendicular to D, remain- ing unchanged. Projecting the other three axes we get x' = J B + ^ C y> = - J B + 1 C *' = - i C. The length of each of these is -^. This length may be found directly by the equation •3xd „ sin - X D = 1 — cos - x D = 1 The variable k^ is independent. The figure is therefore a right prism of unit length along A. To find the prism base, or section in the BC plane, draw the axes kB. hC and plot the figure. First, let k^ = ky = 0, while Ar, varies from to 1, tracing the line along C from — J to O, the line aO. Next let ky = i; the locus of k^is then the line be from B C . _B C ~ 2 6 ^° 2^6- Intermediate values of ky fill out the parallelogram ac. 30 A GiiOMETRICAL VECTOR ALGEBRA Next let k^— 1, and proceed as before, obtaining the parallelogram dC, B C B O whose limiting lines are de from o "" fi *'^ ^ ^ ^ ' ^^^ ^^ from O to IT . Intermediate values of k^ give similar parallelograms commencing at every point along a d and covering the regular hexagon ace. The whole projection is a right hexagonal prism. The projected axes x\ y\ z\ are Oc, Oc, Oa. (4) Parallel to x + y + ar + u, (= D). Take for the other rectangular axes A = X — y B = « — u C = ^ Wabd = x + y — i— t/. /(-, - A-, k,- k, k,^k,-k,- k, Ihen / = n — '- , m = ^ , n = — 7 , and the locus of the projection is 7' 7' 7' V, ^x - k, k, - k, k, ' k,. -k, - /r, where a = b = ^ 2, c" = 2. Projecting the axes x, y, z,u, we get *' = i A + i C . yi = - JA + iC M'=-iB-iC . . 1 3 and the length of each projected axis is -h" . To obtain the geometric form of the projection, give to all the variables the value zero, then to each one separately give all values up to unity. This gives four lines from O, identical with the projected axes. With three of these lines as adjacent edges form a parallelopiped, and form three more parallelopipeds with the three other possible groups of the four lines. The sum of these four solids, a rhombic dodekahedron, is the pi ejection required. :'iii^^. 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. i!iii^" Sneer kH?"' 6i DTL MG-^ im OCTl^l -SGI 40ct'6lJMl HEC'P LB AJul^Q^Q Vr -^sQ^ ^EPir Wr /30rt'6IDM 5Ul5 i Vi-'L-.X^ 23^u^ '62G^y SEP 29 1961 150ct'6lLU REC: MOV ^ «.-i 0G1 '^1 .<<^^'^. \.. U3- LD 21A-50r?i-4 (A1724sl0)47l 2^Ni PEfi^SPW^* LD 21A-60)H-10,'65 (F7763sl0)476B General Library University of California Berkeley eAYLORD BROS, INC. Manufacturen Syracuse, N. Y. Stockton, Calif. U C.BERKELEY UBRARIES CDbl3M'lb33 ^^/gs'S *-i*- ■XG\ H3 THE UNIVERSITY OF CAUFORNIA UBRARY