ax by c 2x y 2 Contains 0 2 Direction 1 2 Normal 2 1 x y 2 2 1 0 Lecture 5 Line in R Parametric Equations Line in R Add a plane 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 a b 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 Direction 1 1 2 Parametric equations x t 1 t y t 2 t z t 3 2t Add plane x 2y z 7 Normal 1 2 1 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 Where is the intersection Two planes intersecting x y z 1 normal 1 1 1 x 2y 0 normal 1 2 0 These intersect to form a line We need a point on the line and a direction Hope line intersects the plane x 0 y z 1 2y 0 0 0 1 is a point on the line Now you could try the plane x 1 Cycloid Parametric Curves in R Circle x t cost y t sint Parabola x t t y t t Another circle x 1 t 1 t y 2t 1 t Wheel rolling with radius 1 with a dot at point p Path of C t starts at 0 1 Direction 1 0 Parameterize CP path of P from the view of C traces circle of radius 1 starts at 0 1 travels clockwise CP sint cost P t C t CP t Instead we will go ahead to try to nd the direction of the line since we know the line is contained in both planes meaning it is perpendicular to both normals Back to the parametric equations x t 0 2t 2t y t 0 t t z t 1 t 1 t
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