Download e-book for kindle: A primer of quaternions - illustrated by Arthur S. Hathaway

By Arthur S. Hathaway

ISBN-10: 1933998644

ISBN-13: 9781933998640

Illustrated, together with a variety of Examples - Chapters: Definitions And Theorems - heart Of Gravity - Curve Tracing, Tangents - Parallel Projection - Step Projection - Definitions And Theorems Of Rotation - Definitions Of flip And Arc Steps - Quaternions - Powers And Roots - illustration Of Vectors - formulation - Equations Of First measure - Scalar Equations, airplane And directly Line - Nonions - Linear Homogeneous pressure - Finite And Null lines - Derived Moduli, Latent Roots - Latent traces And Planes - Conjugate Nonions - Self-Conjugate Nonions - Etc., and so on.

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A sum of quaternions is associative and commutative. CHAPTER 3. QUATERNIONS 35 For since by Cor. , ordinary numbers) is independent of the way in which its terms are added, therefore by Art. 49 the scalar and the vector of a sum are independent of the way in which the sum is added. Hence the sum is independent of the way in which it is added, since it is equal to the sum of its scalar and its vector. 51. Lemma. If p, q be any quaternions, then (1 + p)q = q + pq. For take OB in the intersection of the planes of p, q and draw OA, OC such that q · OA = OB, pOB = OC; then (1 + p)q · OA = (1 + p)OB = OB + OC = qOA + pqOA.

65 take this form by multiplying the first by β, the second by α, and subtracting, remembering that V (V αβ · ρ) = αSβρ − βSαρ. 67. The vector perpendicular distance from the line V δρ − δ = 0 to the point P is δ −1 (V δρ − δ ), where ρ = OP . For the required perpendicular distance of P is the projection of NP , = (ρ − δ −1 δ ), perpendicular to δ. 68. The point of intersection of the three planes Sαρ = a, Sβρ = b, Sγρ = c is [Art. ] ρ= (aV βγ + bV γα + cV αβ) . Sαβγ Examples 1. Find the equation of the locus of a point that moves so that its numerical distances from two fixed points are equal.

Ex. ] (h) Find the angle between two adjacent faces. Chapter 4 Equations of First Degree 61. The general equation of first degree in an unknown vector ρ is of the form, (a) q1 ρr1 + q2 ρr2 + · · · = q, where q, q1 , r1 , q2 , r2 , · · · are known numbers. This equation may be resolved into two equations by taking the scalar and the vector of each member; and we shall consider these equations separately. 62. Taking the scalar of (a), Art. 61, the term Sq1 ρr1 becomes, by a cyclic change in the factors, S · r1 q1 ρ, and this becomes [by dropping the vector (Sr1 q1 )ρ, since its scalar is zero] S(V r1 q1 · ρ); and similarly for the other terms.

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