SOLUTION: 1. Find vectors ~v and ~w such that ~v is parallel to ~u = 2~ı − 3~j, ~w is perpendicular to ~u and ~v + ~w = 4~ı + 4~j. (The signs "~" are arrows that go above) 2. Find al

I will use bold v, u, and w to denote vectors, and

notation like  to denote components of a vector (here x component of vector u)  



Hopefully it renders properly.





u = 2i - 3j           (1)



From v || u: v = 2ci-3cj   (2)

   where  c  is some scaler value.





From w perpendicular to u:  w∙u = 0

   which translates to

         2 - 3 = 0     (3)



We also want:  w + v = 4i + 4j    (4) 



(3) ==>  



Now we can write, using (3) and (4):

 (5)   2c +  = 4

 {6)  -3c +  = 4



Solving this for c gives    which tells us via (2)

that v = (-8/13)i + (12/13)j





Now you can use (5) and (6) to solve for  or plug  c into 

(4) and solve for  and  (I did it the latter way):



 w = (60/13)i + (40/13)j







Check:

I checked everything but you should do so as an exercise.

1.  Check that w + v = 4i + 4j

2.  Check that w∙u = 0  ()

3.  Check that |v∙u| = |v||u|