18.02 Lecture Notes - Lecture 1: Cycloid
Document Summary
Line in r ax + by = c > 2x + y = 2 contains (0, 2) direction <1, -2> normal <2,1> > <2,1> = 0. Parametric equations x(t)=t, y(t)=2-2t > 2x(t) + y(t) = 2t + 2 = 2 all these point are on the line. Given a line through (x , y ) and direction x(t) = x + at, y(t) = y + bt. Q = (1, 2, 3), q = (2, 1, 1), and there is a line through q and q. Parametric equations: x(t) = 1 + t, y(t) = 2-t, z(t) = 3 - 2t. Add plane x + 2y + z = 7. These must intersect unless normal plane dotted with the direction of line equals zero. 7 = x(t) + 2y(t) + z(t) = 1+ t =4 -2t +3 - 2t = -3t + 8 > t=1/3 > (4/3, 5/3, 7/3) is both in the line and the plane.
