Vector Identities and Plane Trigonometry with GA

Previous section: Rotors and Rotations in the Euclidean Plane.

Next sectionModeling 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)    (ab)− (ab)2 a2b2

          (b)    (ab)(bcb2a− (ab)bc

                   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):

a b c c b a α β ϒ



          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 − 2acos γ = c2

          (b)    Law of sines (areas)

\[\Large{\frac{{\sin \alpha }}{a}\,\,\, = \,\,\,\frac{{\sin \beta }}{b}\,\,\, = \,\,\,\frac{{\sin \gamma }}{c}}\]

         
          (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:

a b c a b c a b c aΛc aΛb cΛb


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 sectionRotors and Rotations in the Euclidean Plane.

Next section: Modeling Real Objects and Motions with Vectors.

© David Hestenes 2005, 2014