Math 152 (TTH)
Spring 2013
Exam 1
Feb 7, 2013
Time Limit: 50 Minutes
Name (Print):
Student number
This exam contains 8 pages (including this cover page) and 5 problems. Check to see if any
pages are missing. Enter all requested information on the top of this page, and put your
initials on the top of every page, in case the pages become separated.
You may not use your books, notes, or any calculator on this exam.
You are required to show your work on each problem on this exam. The following rules apply:
• Organize your work, in a reasonably neat
and coherent way, in the space provided. Work
scattered all over the page without a clear ordering will receive very little credit.
Problem Points Score
1
8
• Mysterious or unsupported answers will
not receive full credit. A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect
answer supported by substantially correct calculations and explanations might still receive
partial credit.
2
5
3
3
4
5
5
4
• If you need more space, use the back of the
pages; clearly indicate when you have done
this.
Total:
25
The maximum grade you can get is 25. You may use
the table to the right to get an idea of the weight of
each question, but please don’t write in it.
Math 152 (TTH)
Exam 1 - Page 2 of 8
I. Short Answer Problems
1. In problems (A)–(D), let ~u = [1, −2, 3], ~v = [3, 4, 0] and w
~ = [−1, 5, 2].
(A) (1 point) Calculate 3~u − 2w.
~
(B) (1 point) Calculate k~v k.
(C) (1 point) Calculate ~u × ~v .
(D) (1 point) Are the vectors ~u, ~v , and w
~ linearly independent?
Feb 7, 2013
Math 152 (TTH)
Exam 1 - Page 3 of 8
Feb 7, 2013
In problems (E)–(F), let ~a, ~b and ~c be non-zero vectors in R3 . For each statement,
determine whether it is true or false. If true, give a brief explanation. If false, give a
specific example for which the statement does not hold.
(E) (1 point) If ~a · ~b = 0 then ~a × ~b 6= ~0.
(F) (1 point) If {~a, ~b}, {~b, ~c}, and {~a, ~c} are each linearly dependent, then {~a, ~b, ~c} is
also linearly dependent.
Math 152 (TTH)
Exam 1 - Page 4 of 8
Feb 7, 2013
(G) (1 point) Consider the planes 3x + 2y − z = 5 and ax − 6y + bz = 2. For what
values of a and b are the planes parallel to each other?
(H) (1 point) Write a MATLAB for loop that produces

2 3 4 5 ···
 3 4 5 6 ···

A =  ..
 .
11 12 13 14 · · ·
the following 10 × 20 matrix:

21
12 

.. 
. 
30
Math 152 (TTH)
Exam 1 - Page 5 of 8
Feb 7, 2013
II. Long Answer Problems
2. Consider the vectors ~v = [1, −2, 3] and ~u = [2, 1, 5].
(a) (3 points) Find a parametric description of the line through the point (−1, 1, 3)
orthogonal to both vectors ~v and ~u.
(b) (2 points) Find the equation of a plane through the point (−1, 1, 3) orthogonal to
the vector ~u.
Math 152 (TTH)
Exam 1 - Page 6 of 8
3. (3 points) Consider the two planes given by the equations
2x + y + z = 6
and
−4x − 2y − 2z = 24.
Find the distance between these planes.
Feb 7, 2013
Math 152 (TTH)
Exam 1 - Page 7 of 8
Feb 7, 2013
4. (5 points) Find the value of h so that d~ = [1, −6, 4, h] is a linear combination of ~a =
[1, −2, 1, −5], ~b = [0, −4, −1, 1], and ~c = [0, 0, 2, 3].
Math 152 (TTH)
Exam 1 - Page 8 of 8
Feb 7, 2013
5. (4 points) The reduced row echelon form of the augmented matrix for a linear system is


1 0 0 0 0 1
 0 1 2 3 0 2 


 0 0 0 0 1 1 .
0 0 0 0 0 3
(a) What is the rank of this augmented matrix?
(b) Specify whether this linear system has a unique solution, infinitely many solutions,
or no solution. Justify your answer.
(c) Find all solutions (if any) of the corresponding homogeneous system.