High School Geometry with Geometric Algebra

Previous section: Modeling Real Objects and Motions with Vectors.

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Nonparametric equation for a line {x} through point a with direction u:    (x a u = 0. 


Exercises:

          (1)  Sketch the line. 

          (2)  Derive an equivalent parametric equation for the line with  u2 1.

          (3)  Find the directed distance d from the origin 0 to the line and sketch. 

          (4)  Find the directed distance from an arbitrary point y to the line.


Hints:

          (2)  Write  (xa)= α  and solve for  x αa.

          (3)  xadu. Sketch the directed areas and solve for d


(
Click here for solutions to exercises on this page.)



Rigid displacements in a plane: Congruence and measurement

y' r' x y r θ O a O' x'= R x+a R x



Equations for a rigid displacement of particles in a body or points in a reference frame:

          x x′ = Ra                    eiθ          R eiθ

          y y′ = Ra          


Rigid displacement of an interval:

          r = x y    →    r′ = x′ − y′ = R(x y)

or

          r′ = Rr = rR 


Invariants of rigid displacements:  Euclidean distance:    (x′ − y)2 (− y)2



Previous sectionModeling Real Objects and Motions with Vectors.

Next section: Basic Kinematic Models of Particle Motion.


© David Hestenes 2005, 2014