Download A primer of quaternions - illustrated by Arthur S. Hathaway PDF

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, airplane And instantly Line - Nonions - Linear Homogeneous pressure - Finite And Null traces - Derived Moduli, Latent Roots - Latent traces And Planes - Conjugate Nonions - Self-Conjugate Nonions - Etc., and so on.

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1 1 6 (a) Find the values of [(2, 50◦ )6 ] 3 , [(2, 50◦ ) 3 ]6 , and (2, 50◦ ) 3 . 4. What numbers are represented by 2 feet, 2 feet east, the unit of length being a foot, a yard, an inch? 5. Show that i2 = j2 = k2 = ijk = −1; jk = i = −kj; ki = j = −ik; ij = k = −ji. 6. Let e(AB) denote the versor that turns counter-clockwise round the axis AB through an arc that is formed by bending the length AB into an arc of unit radius. Show that if facing the west, and holding the paper in a north and south vertical plane, then ei , e2i , · · ·e−i , e−2i , turn respectively 1, 2, · · · radians counter-clockwise, and 1, 2, · · · radians clockwise in the plane of the paper.

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.

54. (a) T (· · · pqr) = · · · T p · T q · T r. (b) U (· · · pqr) = · · · U p · U q · U r. (c) ∠(· · · pqr) = ∠(r · · · pq) = ∠(qr · · · p), etc. (d) S(· · · pqr) = S(r · · · pq) = S(qr · · · p), etc. (e) T V (· · · pqr) = T V (r · · · pq) = T V (qr · · · p), etc. (f) arc (· · · pqr) = arc r + arc q + arc p + · · · . (g) (· · · pqr)−1 = r−1 q −1 p−1 · · · . (h) K(· · · pqr) = Kr · Kq · Kp · · · . CHAPTER 3. QUATERNIONS 37 (i) S(xp + yq + zr) = xSp + ySq + zSr, [x, y, z, scalars] and similarly for V or K instead of S.

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