## 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??

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 − 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:

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.