Another service from Omega

Vectors in 3D

You may need the mvcal package
Here are some exercises to be done with MAPLE.
We'll go over some of them in class or recitation.

Module 03 -- WorkSheet     Name : ________________________________

1) 	a)  Make up a vector v. Rewrite in the coordinates form ( , , ).
	b)  Store the vector in Maple.

> with (linalg) ; 
> v := vector( [         ,        ,        ] ) ;    

#(Type in the coordinates of the vector  )
	c)  Make up another vector w. Make sure is not a scalar multiple of.
Rewrite w in the form (       ,        ,       ).  Then store it in Maple.

> w := vector( [         ,        ,        ]  ) ;

	#d)  Now calculate the following by hand :  

	v + w = ______________               5 v - 3 w =  ________________
	e)  Check your calculation with Maple :
> evalm (v + w) ;
> evalm ( 5*v - 3*w ) ;

	#f)  Calculate (by hand)  v dot w = _______________.
	g)  Check your answer with Maple :
> dotprod (v, w) ; 

	#h)  Calculate (by hand) the cosine of the angle between v and w :  
		cos(theta) = ______________
                                    

Tell from this answer if the angle between the two vectors is less than, equal
to or greater than Pi/2 .  
         Ans :   ______________________           Explain : _______________
	i)  To find the exact value of theta:
> evalf( arccos(                            ) );
#To check your calculation with Maple :
> evalf( angle (v, w) ); 

#2) 	a) Make up two vectors :  v, w. Make sure that w is not a scalar
multiple of v.  
> with (linalg) ;   
>   v := vector( [         ,        ,        ] ) ;  w := vector([         ,    
   ,        ]) ;

#The commands :
> with( mvcal) :  arrows( { [    ,     ,   ] , [     ,   ,    ] } ); 

#will show the vectors  and .  
	b) Calculate (by hand) the vector "projection of v onto w" =  __.
	c) Check your answer in b) with Maple :

> proj := evalm( dotprod(v, w)/dotprod(w,w) * w );

	#d) Is it true that proj. v on w = proj. w on v?   
	     Ans : _____________,  Explain  ____________________
	e) When we calculate (v - proj(v,w)) dot w , the answer is 0, see :

> dotprod (  (v -proj), w ) ;
	#Can you explain why ?     

Reason _____________________________________________________

3) 	Use the vectors  and  in the last two problems. 

> v := vector( [     ,     ,    ] ) ;  w := vector( [     ,     ,    ] ) ;

	#a) Calculate  v cross w by hand.  Let us call this vector u.
 u =  ________________
	b) Check your calculation with :

>  u := crossprod( v, w) ;
#To see the picture of ,  and , you can type in the coefficients of  and  :
> crossproduct( [    ,    ,    ],  [   ,    ,    ] ,init = [0 ,0, 0] ) ;
	#c)The last vector (0,0,0) in the previous command specifies 
		the staring point of all the vectors.  If we choose a 
		different starting point, say ( 1,-2, 1), then u will be the 
		vector joining (1, -2, 1) to which point ?        
Ans  ____________________.   Check your answer from the picture :

> crossproduct( [    ,    ,    ],  [   ,    ,    ] , init = [1 ,-2, 1] ) ;

	#d)  Now, try the following calculations using Maple :
>  dotprod( v, u) ; dotprod( w, u) ; dotprod( 0.21*v + 0.5*w, u) ; 
>  dotprod(100*v-0.6*w, u) ;

Can you explain why all the answers in the above calculations are zero ? 
(Remember u = v x w.)

Reason : ___________________________________________________________