Worksheet to Support Powerpoint Presentation

Tangent Vectors, Normal Vectors, Acceleration, Curvature

This worksheet was used to create some of the pictures in the powerpoint presentation on tangent vectors, normal vectors, acceleration, and curvature.

>	with(plots):

Here is the curve all by itself.

>	spacecurve([t,6-t^2,t,t=-2..0],thickness=2,color=blue,axes=normal,orientation=[40,75],labels=[x,y,z]);

Here is the curve along with the acceleration vector in red, principal unit tangent vector in green, and principal unit normal vector in magenta.  They are all at the point where t = -1, the point (-1,5,-1).

>	curve:=spacecurve([t,6-t^2,t,t=-2..0],thickness=2,color=blue,axes=boxed,orientation=[40,75],labels=[x,y,z]):

>	Avector:=arrow(<-1,5,-1>,<0,-2,0>,width=[0.02,relative],color=red,thickness=3):

>	Tvector:=arrow(<-1,5,-1>,<1/sqrt(6),2/sqrt(6),1/sqrt(6)>,width=[0.05,relative],color=green,thickness=3):

>	Nvector:=arrow(<-1,5,-1>,<-1/sqrt(3),1/sqrt(3),-1/sqrt(3)>,width=[0.05,relative],color=magenta,thickness=3):

>	display(curve,Avector,Tvector,Nvector);

The graph below duplicates the graph above except the scaling has been constrained so that the tangent vector and normal vector do look orthogonal.

>	curve:=spacecurve([t,6-t^2,t,t=-2..0],thickness=2,color=blue,axes=boxed,orientation=[40,75],labels=[x,y,z],scaling=constrained):

>	display(curve,Avector,Tvector,Nvector);

Here is the curve in the plane along with the principal unit tangent vector in green at the point (1,5).

>	curve1:=plot([t,6-(t-2)^2,t=0..2],color=blue,thickness=2):

>	curve2:=plot([t,5,t=0..2],color=navy,linestyle=DASH):

>	vector1:=arrow(<1,5>,<1/sqrt(5),2/sqrt(5)>,color=green,width=[0.02,relative]):

>	display(curve1,curve2,vector1);

Here is a series of principal unit tangent vectors along the same curve.

>	curve3:=plot([t,15/4,t=0..2],color=navy,linestyle=DASH):

>	vector2:=arrow(<1/2,15/4>,<1/sqrt(10),3/sqrt(10)>,color=green,width=[0.02,relative]):

>	curve4:=plot([t,23/4,t=0..2],color=navy,linestyle=DASH):

>	vector3:=arrow(<3/2,23/4>,<1/sqrt(2),1/sqrt(2)>,color=green,width=[0.02,relative]):

>	curve5:=plot([t,2,t=0..2],color=navy,linestyle=DASH):

>	vector4:=arrow(<0,2>,<1/sqrt(17),4/sqrt(17)>,color=green,width=[0.02,relative]):

>	display(curve1,curve2,curve3,curve4,curve5,vector1,vector2,vector3,vector4);

Here is the same curve but we see a little more of it, one additional principal unit normal vector, and the same scale is being used on each axis (scaling = constrained).

>	curve1:=plot([t,6-(t-2)^2,t=0..3],color=blue,thickness=2,scaling=constrained):

>	curve6:=plot([t,6,t=0..2],color=navy,linestyle=DASH):

>	vector5:=arrow(<2,6>,<1,0>,color=green,width=[0.02,relative]):

>	display(curve1,curve2,curve3,curve4,curve5,curve6,vector1,vector2,vector3,vector4,vector5);

>