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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 (x−a)∙u = α and solve for x = αu + a.
(3) x∧u = a∧u = du. 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
Equations for a rigid displacement of particles in a body or points in a reference frame:
x → x′ = Rx + a R = eiθ R✝ = e−iθ
y → y′ = Ry + a
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 = (x − y)2
Previous section: Modeling Real Objects and Motions with Vectors.
Next section: Basic Kinematic Models of Particle Motion.