By Arthur S. Hathaway
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, aircraft And instantly 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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Extra resources for A Primer of Quaternions - Illustrated
For Exercise and Reference 53. (a) q = T q · U q = U q · T q. (b) q = Sq + V q; Kq = Sq − V q. (c) Sq = T q cos ∠q, = r cos θ, say; T V q = T q sin ∠q = r sin θ. (d) V q = T V q · U V q = r sin θ where = U V q. (e) q = r(cos θ + · sin θ) = re , Kq = r(cos θ − sin sin θ) = re−θ θ (f) eθ · eθ = e(θ+θ ) . (g) Sq = 12 (q + Kq); V q = 21 (q − Kq). (h) T q 2 = qKq = Kq · q = (Sq)2 − (V q)2 = (Sq)2 + (T V q)2 . (i) q −1 = Kq/T q 2 . As a further exercise find the T , U , S, V , K of the T , U , S, V , K of q, in terms of r, θ, .
57. (a) i2 = j2 = k2 = ijk = −1; jk = i = −kj; ki = j = −ik, ij = k = −ji. (b) ρ = iSiρ − jSjρ − kSkρ. ] Let ρ = xi + yj + zk, ρ = x i + y j + z k, etc. [x, y, z, etc. ] Then, prove by direct multiplication, (c) −ρ2 = x2 + y 2 + z 2 = T ρ2 . (d) −Sρρ = xx + yy + zz = −sρ ρ. (e) vρρ = y y z z i+ z z x (f) −Sρρ ρ = x x y y y x x j+ x x y k = −V ρ ρ. y z z = −SρV ρ ρ . z Geometric Theorems 58. The angle of αβ equals the supplement of the angle θ between α, β. CHAPTER 3. QUATERNIONS 39 For, since αβ · β −1 = α, therefore αβ turns through the angle from β −1 to α, which is the supplement of the angle θ from α to β.
14, 15, 16. 18. The equation of the plane through AB that contains the common perpendicular to AB, CD is S · APV (ABV ABCD) = 0, or 2x + y + 2z = 32. 19. The equation of the line through A parallel to OD is V ODAP = 0 or AP = = (y−2) = (z−7) . tOD, or (x−8) 9 10 11 20. The equation of the line AB is V ABAP = 0, or AP = tAB or (z−7) . −4 (x−8) 2 = (y−2) 4 = 21. The equation of the common perpendicular to AB, CD is the equation of Ex. 18 and x + 2y − 2z = 7. 22. Find the distance of D from the lines in Exs.
A Primer of Quaternions - Illustrated by Arthur S. Hathaway