Vectors in 3D

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 : ___________________________________________________________