Homework 2, 6/28/05 At the end of yesterday's session the question came up of how to add a normal line to a plane. We start with our normal set-up restart: with(LinearAlgebra): with(plots): with(plottools):

Warning, the name changecoords has been redefined

Warning, the assigned name arrow now has a global binding

We then find the matrix and the set of point like last time. A := RandomMatrix(1,3,generator=rand(-10..10)); A := <A,rand(-10..10)()*Row(A,1)>; setofpoints := {seq(Vector(3,[rand(-10..10)(),rand(-10..10)(),rand(-10..10)()]), i=1..100)}: pointplot3d(setofpoints,view=[-10..10,-10..10,-10..10], axes=normal,color=black):

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Last time we plotted the image of our 100 random points and found a basis of the null space. setofimages := {seq(A.setofpoints[i],i=1..100)} minus {0}: pointplot(setofimages,color=black); nullbasis:=NullSpace(A);

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We then plotted linear combinations of the basis vectors of the null space. nullpoints := {seq(rand(-10..10)()*nullbasis[1] + rand(-10..10)()*nullbasis[2],i=1..500)} minus {0}: pointplot3d(nullpoints,view=[-10..10,-10..10,-10..10], axes=normal,color=black);

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We want to take a cross product to get a normal vector. v1 := nullbasis[1]; v2 := nullbasis[2]; v3 := CrossProduct(v1,v2);

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Normalize the normal vector. V3 := v3/sqrt(v3.v3);

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Now we make the plots of the points and the vector into separate plot structurse and display them together. SC := spacecurve(V3*t, t=0..3, thickness = 5,color=blue): PP := pointplot3d(nullpoints,view=[-10..10,-10..10,-10..10], axes=normal,color=black): display3d(SC,PP);

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