Another service from Omega

Lines and Planes in 3D

You may need the mvcal package
NOTE: To be able to display 3D and high resolution graphics you
need to use a powerful enough terminal or screen (e.g. IBM-PCs
running windows, UNIX workstations running Xwindows, Macs., or
dial-up terminals with TEKTRONIX emulation)

Module 04 -- Work Sheet

1)
 	a)  Make up a vector v = _i+ _j+ _k  and a point P=(  ,  ,  ). 
Write down an equation of a line L1, that passes through the point P
and has direction v.  

L1(t) = _______i + ______j  + _______k  =  (     ,    ,       )

To see the picture of this line :

>  L1 := plot3d ( [     ,      ,      ], t = -2..2, s =-1..1,grid = [2,2]) :
>  with (plots) ;  display3d ({L1} ) ; 

	#b)  Write down an equation of the line L2 that passes through P
and has direction  2v.

L2(t) = (        ,        ,        )

>  L2 := plot3d ( [      ,     ,   ], t = -2..2, s =-1..1,grid = [2,2]) ;
>  with (plots) ;  display3d ({L2} ) ; 

	#c)  Now try the following command :
>  with (plots) ;  display3d ({L1, L2} ) ; 
	#This command is supposed to show the lines L1 and L2,
how come only one line is shown in the picture ? 

 Explain :        __________________________________________


2)
	a)  Make up a vector v = (  ,   ,  ) which is not parallel to any of
the axes.  Make up a point P =  (  ,  , ).  Write down the equation of 
the plane with normal v which passes through P.  

Ans :   _____x+________y+________z = ________ .

Rewrite it in the form : 

	z = _______x+_________y+________
	 
We need this form to plot a picture of the plane :

>  plot3d (          ,  x =-2..2, y = -2..2) ;

	#c)  Write down the equations of 3 other planes that are 
parallel to the plane in part a). 

Ans :   __________,    ___________,   __________,

Rewrite them in the form :  z = (something)

Plot the 4 planes together, to see if they are parallel :

>  plot3d (  {     ,    ,   ,    } ,  x =-2..2, y= -2..2) ; 

	#c)  Write down the formula for the xy plane : z=_______
What is the normal vector of the xy plane?.  Ans :  (  ,   ,  ).
Find the angle between the normal vector v (in part a) and the normal 
vector of the xy plane.  
	       Ans :    Angle  =  arccos(            )  
	d)  What is the angle between the plane in part a) and the xy plane?. 
	       Ans :    Angle  =   arccos(            ) 


3)	Let  L1 be the line : x=t+3, y=2t-1, z=5t+1.
To see its picture, you can either use :

>  spacecurve( [t+3, 2*t -1, 5*t+1], t=0..1) ;   

or 
>  plot3d( [t+3, 2*t-1, 5*t+1], t=0..1, s=0..1, grid=[2,2]);

	#a)  Write down the direction of L1, and find a point that 
it passes through.

Direction : _______________            A point : ___________________

	b) Find an equation of a line which will intersect L1.

Ans : x=__________, y=____________, z=_____________

Rewrite the equation using s as the parameter instead of using `t' :

x=______________,  y=___________,  z=_______________

You can check if your line really intersects with  by using Maple :

>  solve( {x = t + 3 , y =  2*t -1 , z =5*t+1, x =    ,  y =  , 
z =     }, {x, y, z,s , t } ) ;

	#c)  Which of the following planes will contain the line L1 ?
		z=x+y+2,   z=x+2y.
 
Explain :  ____________________________________________________

>  line := plot3d([t+3, 2*t-1, 5*t+1], t=0..1, s=0..1, grid=[2,2]) :
>  plane1 := plot3d( x + y + 2, x=-2..2, y=-2..2) :
>  plane2 := plot3d( x + 2*y, x=-2..2, y=-2..2) :
>  display3d( {line, plane1}, style = WIREFRAME) ;
>  display3d( {line, plane2}, style = WIREFRAME) ;