Previous section: Rotors and Rotations in the Euclidean Plane.
Next section: Modeling Real Objects and Motions with Vectors.
Exercise 1: Prove the following vector identities and show that they are equivalent to trig identities for unit vectors
in a common plane.
(a) (a∙b)2 − (a∧b)2 = a2b2
(b) (a∧b)(b∧c) = b2a∙c − (a∙b)b∙c
Hint: Expand abbc in two different ways.
Exercise 2: Use rotor products to derive the trigonometric "sum of two angles" formulas:
(a) cos(θ+φ) = ??
(b) sin(θ+φ) = ??
(Click here for solutions to exercises on this page.)
Plane (Euclidean) Trigonometry can be reduced to two basic problems:
(a) Solving a triangle
(b) Solving the circle (composition of rotations).
A triangle relates 6 scalars: 3 sides (S) & 3 angles (A):
Given 3 of these scalars (SSS, SAS, SSA, ASA, AAS, AAA),
solving the triangle consists of determining the other 3 scalars.
Laws of the triangle follow directly from the geometric product:
(a) Law of cosines (scaled projection): a2 + b2 − 2ab cos γ = c2
(b) Law of sines (areas)
(c) Angle laws:
• Supplementary angles (interior/exterior)
• Sum of interior angles
• Sum of exterior angles
Exercise 3: Derive these laws with GA and interpret with diagrams, such as the following:
Is it obvious that the green, yellow, and blue areas are all the same and equal to twice the triangle's area? If not, you can prove that by starting with c = a + b on your way to proving the law of sines.
(Click here for solutions to exercises on this page.)
Previous section: Rotors and Rotations in the Euclidean Plane.
Next section: Modeling Real Objects and Motions with Vectors.