EXAM EM&M

WRITTEN EXAM EM&M
NOVEMBER 6TH, 2015
INSTRUCTIONS
This written exam is 60% of the total grade of EM&M.

In total you can earn 100 points. When you earned 55 points or more, you will have passed this
written exam.

Grade = number of earned points/100

Please formulate your answers in English.

Always show your work.

When you get stuck at a question, do not linger to long. First try do the questions you do know the
answer to.

Good luck!


Question 1 - Vectors (20 points)
      
If: 
p = 2 i -− 4 j + 3 k , 
q = 4 i + j + 12 k and 
r = 3 i -− k
a) Find: 3 p +2 q
b) Find: q x 
 r
c) Show that  q and r are perpendicular
d) Define the unit vector in the direction of  p

solution:
           
a) 3 2 i -− 4 j + 3 k  + 2 4 i + j + 12 k  = 6 i -− 12 j + 9 k  + 8 i + 2 j + 24 k 
  
= 14 i -− 10 j + 33 k (5 points, matrixform also correct, miscalculation = 0 points)

      
b) 2 i -− 4 j + 3 k  x3 i -− k  = (-−1 -− 0) i + (36 + 4) j + (0 -− 3) k
  
= -− i + 40 j -− 3 k
(5 points, matrixform also correct, 1 miscalculation = 3 points, more miscalculations =
0 points)

c) 
q ·
r = 12 + 0 -− 12 = 0 (5 points)
so 
q and  r are perpendicular

     

p 2 i -−4 j +3 k 2 i -−4 j +3 k 2  4  3 
d)  = = = i -− j+ k (5 points)
p 4+16+9 29 29 29 29

, 2|




Question 2 - Matrices (20 points)
-−3 5 -−1 3 4 -−2 2
If A = ; B= ; C=
7 -−4 2 1 -−1 3 1

Calculate, if possible:
a) A - 4B
b) CT A
c) CB
d) det(B)

solution:
-−3 5 -−1 3 -−3 5 -−4 12 1 -−7
a) -− 4 = -− =
7 -−4 2 1 7 -−4 8 4 -−1 -−8
(5 points, 1 miscalculation = 3 points, more miscalculations = 0 points)

4 -−1 -−19 24
-−3 5
b) -−2 3 · = 27 -−22
7 -−4
2 1 1 6
(5 points, 1 miscalculation = 3 points, more miscalculations = 1 point if CT is correct
otherwise 0 points)

c) Not possible (5 points)

d) -−1·1 -− 3·2 = -−7 (5 points)