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) ;