# 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 = _______________.
> 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" =  __.

> 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 ?

> 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 : ___________________________________________________________

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