MATH 152 PRACTICE PROBLEMS FOR MIDTERM 1
(1) Find the value of cos θ where θ is the angle between the vectors
~a = (1 , 2 , 9)
~b
= ( − 1 , 4 , 1)
(2) Find the area of the triangle formed by the three points
(1 , 2 , 1) (0 , 5 , 1) (2 , 3 , 7)
(3) If (1 , 1 , 1) and (2 , 3 , 7) are points on a line, determine a parametric form of the line.
(4) Find an equation form of the line in problem 3.
(5) The points (1 , 1 , 1), (2 , 3 , 7), and (0 , 2 , 0) are on a plane. Write a parametric form and an equation form for the plane.
(6) Find the intersection of this plane x − 2 y + 3 z = 2 with the line
~ = (1 , 0 , 0) + t (1 , 2 , 1)
(7) Find the closest point on the plane x
1
+ x
2
+ x
3
= 1 to the point (1 , 2 , 0).
(8) Consider the lines
L
1
: s (1 , 1 , 1)
L
2
: (1 , 0 , 0) + t (1 , 2 , 3)
For what point on L
1 and what point on L
2 is the distance between the two lines as small as possible? Hint: (See figure 1) This occurs when the vector ~ shown is orthogonal to both the line directions of L
1 and L
2
.
(9) Find the value of a so that the matrix

1 1


2 a
 has determinant 0.
1

2 MATH 152 PRACTICE PROBLEMS FOR MIDTERM 1
Figure 1.
The shortest vector between lines L
1
, L
2
(10) For what value (or values) of a , if any, does the matrix
 a 1 1

 


1 1 2 


2 1 3
 have determinant 0?
(11) Write a parametric form of the plane x
1
+ x
2
+ x
3
= 1
(12) For what values of a , if any, is the set of vectors
(1 , 1 , 1) , (1 , 2 , 1) , ( a, 2 , 3) linearly independent? Do this question in two ways: using the determinant, and also put the vectors in the columns of a matrix and perform Gaussian elimination.
(13) Find the solution (or solutions) of the following system of equations, if any.
x
1
+ x
2
+ x
3
= 2
2 x
1
+ x
2
+ 2 x
3
= 5
3 x
1
+ x
2
+ 2 x
3
= 6

MATH 152 PRACTICE PROBLEMS FOR MIDTERM 1 3
(14) The system x + y + 2 z = 2
2 x + z = 5 represents a line. Find a parametric form of the line. Do this in two ways. First, “guess” a point on the line and find the line direction using the cross product. Second, by doing
Gaussian elimination and substituting back into the system.
(15) The following linear systems have been put in echelon form. For each system determine
• if the system is in reduced row echelon form
• if the system has no solutions, exactly one solution, or an infinite number of solutions.
In this last case, determine the number of parameters (the dimension) in the set of solutions.
   
1 1 1 1 2 3

1 5 0 2 3
 a)




0 0 1



 b)




0 1 5



 c)

0 0 1 3 2
0 0 0 0 0 0

  d)


1 0 2
0 1 3

 e)

1 0 0 1




0 1 0 2
0 0 1 3



 f)

1 0 2 3




0 5 3 2
0 0 9 9



 g)

1 0 0







0 1 0
0 0 1
0 0 0





 h)

1 2 5 0

0 3 6 0


(16) For the systems in the previous exercise that had solutions, find all solutions.
(17) For the systems in exercise 15 that were not in reduced row echelon form, put them in reduced row echelon form.
(18) Consider the system of equations written as an augmented matrix

1 1 1 0




1 2 5 b
1 3 a 1





For what value (or values) of a and b , if any, does the system have
• no solutions,
• exactly one solution,
• an infinite number of solutions?

4 MATH 152 PRACTICE PROBLEMS FOR MIDTERM 1
It is not necessary to find the solutions when the exist.
(19) Perform Gaussian elimination on the matrix

1 1 5 2

 


1 3 2 1 


2 2 4 0
 to put it in echelon form. Find all solutions of the homogeneous system with coefficients in this matrix.