By Arthur S. Hathaway
Illustrated, together with quite a few Examples - Chapters: Definitions And Theorems - middle 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 instantly Line - Nonions - Linear Homogeneous pressure - Finite And Null lines - Derived Moduli, Latent Roots - Latent strains And Planes - Conjugate Nonions - Self-Conjugate Nonions - Etc., and so forth.
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Extra info for A primer of quaternions - illustrated
CHAPTER 3. QUATERNIONS 42 11. If arc q = BA, arc p = AC, show that the last equation of Ex. 10 is the property cos a = cos b cos c + sin b sin c cos A of the spherical triangle ABC. ] 12. If α, β, γ, α , β , γ are vectors from O to the vertices A, B, C, A , B , C of two spherical triangles on the unit sphere O, where α = U V βγ, β = U V γα, γ = U V (αβ); then α = U V β γ , β = U V γ α , γ = U V α β , and the two triangles are polar triangles. 13. In Ex. ; cos A = S(U V γα · U V αβ) = Sβ γ = − cos α , etc.
Those lines and planes that remain unaltered in geometrical position by the strain φ are called latent lines and planes of φ. (a) The latent directions of φ are the null directions of φ − g1 , φ − g2 , φ − g3 , and g1 , g2 , g3 are the corresponding ratios of extension in those directions. For if ρ is any latent direction, and g is the ratio of extension in that direction, then we have φρ = gρ or (φ − g)ρ = 0. Hence φ − g is a null nonion, or mod (φ − g) = 0, so that g is a latent root of φ; also ρ is a null direction of φ − g.
By ordinary algebra, there are n such nth roots, real and imaginary. , roots of the form a + b −1, where a, b are real quaternions. Representation of Vectors 44. Bold-face letters will be used as symbols of vectors only. In particular, i, j, k will denote unit vectors whose axes are respectively a unit length east, a unit length north, and a unit length up. More generally we shall use the step AB to denote the vector whose axis is a unit length in the direction of AB, and whose tensor is the numerical length of AB (= AB: unit length).
A primer of quaternions - illustrated by Arthur S. Hathaway