ENG EK 103 FALL 2026: Computa6onal Linear Algebra: Problem set 1
For all homework this semester, you should:
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Write down your name, BU ID, and homework number (PS1) at the top of the page
Clearly indicate each problem number for accurate grading
Describe, step by-step, derivaGons and reasoning behind your answers. Answers without jusGficaGons will
receive no credit
Problems that do not involve MATLAB should be solved by hand. However, you may use MATLAB to check
your own answers.
Please note that in EK103 problem sets, a linear-algebra vector will be denoted (as in our textbook) by a
bold lower-case leUer (e.g., 𝒗) OR by a non-bolded lower-case leUer with an arrow or harpoon on top (e.g.,
𝑣⃑). On the other hand, a physics vector (such as one represenGng displacement of an object in space) from
a point 𝐴 to a point 𝐵 will be denoted by &&&&&⃑
𝐴𝐵.
Problem 1:
Suppose you were taking Physics 1 (PY 211) this semester, and you were given a situaGon where a triangular mass
point D is being suspended by a series of pulleys, and its weight is being counterbalanced by a semicircular mass
located at the origin 𝑂 (0,0). This is depicted in Figure 1a below.
Engineering applicaGon: If you were going on a rock-climbing trip in New Hampshire, you (the triangular mass
equivalent) will be suspended by a pulley system consisGng of a rope that’s supported by rock bolts and carabiners.
The physics will be the same!! See Figure 1b for comparison.
Figure 1b: A rock climber will use the same
Figure 1a: A mass-pulley system
&&&&&⃑ and &&&&&⃑
a) Find the vectors 𝐴𝑂
𝐴𝐵 and calculate their lengths.
pulley system!!
**For each vector, be careful on the locaGon of its tail (starGng point) and its Gp (endpoint).
b) Using dot products, find 𝑐𝑜𝑠(𝜃1). Then, using your scienGfic calculator calculate angle 𝜃1.
&&&&&⃑, prove that angle 𝜃2 is a right angle (i.e. , 𝜃2 = 900).
c) Using dot products and the lengths of vectors &&&&&⃑
𝐵𝐴 and 𝐵𝐶
Show all work.
d) Using dot products and the lengths of vectors 𝐶𝐵 and 𝐶𝐷, calculate angle 𝜃3. Show all work.
e) Find the total length of the red rope from point 𝑂 to point 𝐷. Use your calculator and express your answer in
decimals.
Hint: Don’t reinvent the wheel and try to re-calculate each rope segment’s length using Pythagorean Theorem. This
is because by now, you should have the lengths of all the vectors that would define the enEre path of travel of
the red rope.
Problem 2:
−1
Throughout this problem, let 𝑣⃑ = 5 2 9 represent a vector whose Gp (head) in a (𝑥! , 𝑥" , 𝑥# ) coordinate system has the
1
coordinates (-1,2,1).
a) Determine ||𝑣⃑||, the length of the vector 𝑣⃑.
b) What are the coordinates of the Gp of the vector -2𝑣⃑ and specify its length.
2
c) Suppose we added the vector 𝑟⃑ = 509 to the vector 𝑣⃑. What is the length of the resultant vector?
0
d) Determine a non-zero vector 𝑤
&&⃑ (at least one element of 𝑤
&&⃑ must be non-zero) such that the dot product 𝑣⃑ ∙ 𝑤
&&⃑ =0.
Are there other possibiliGes for the selecGon of the vector 𝑤
&&⃑ such that 𝑣⃑ ∙ 𝑤
&&⃑ =0? If your answer is yes, give an
example. If your answer is no, explain in a sentence or two.
e) Determine a vector 𝑢
&⃑ such that the dot product 𝑢
&⃑ ∙ 𝑣⃑ = 1. Are there other possibiliGes for the vector 𝑢
&⃑ such that
𝑢
&⃑ ∙ 𝑣⃑ =1? If your answer is yes, give an example. If your answer is no, explain in a sentence or two.
f) Determine a 3x3 matrix 𝑀 that has at least one non-zero element such that the product 𝑀𝑣⃑ = 0. Hint: The
product 𝑀𝑣⃑ can be conceptualized as a set of three dot products. Are there other possibiliGes for the matrix 𝑀
such that 𝑀𝑣⃑ = 0? If your answer is yes, give an example. If your answer is no, explain in a sentence or two.
Problem 3:
In this problem, you are given that:
1 0 0
1 2
0 2 09, where 𝑞 is another unspecified real number,
𝐴=A
,
where
𝑝
is
an
unspecified
real
number,
𝐵
=
5
C
−1 𝑝
0 0 𝑞
1 2 3
1 2 3
3
5
𝐶 = 53 2 19, 𝐷 = 56 4 29, 𝑥⃑ = H I, and 𝑦⃑ = H I
1
0
2 1 3
6 3 9
(a) Use the equaGon 𝐴𝑥⃑ = 𝑦⃑ to solve for the numerical value of 𝑝. Make sure to clearly show the steps you took to solve for
𝑝.
(b) Use the equaGon 𝐵𝐶 = 𝐷 to solve for the numerical value of 𝑞. Make sure to clearly show the steps you took to solve for
𝑞.
(c) Is it possible to determine the product 𝐷𝑥⃑? If your answer is yes, compute the product. If your answer is no, explain why
not.
(d) Determine the transpose of the matrix D.
Problem 4:
Download and install MATLAB before beginning this problem. The syllabus has instrucGons. This course does not support the
use of MATLAB from your web browser; you must have MATLAB running as a program on your computer.
Download the file ps1_ek103_f26.m from Slack. Move it to an appropriate folder on your computer, such as
Documents/Fall 2026/EK 103/PS1 . Do not try to "run" the file from Slack (that is not possible), and do not try to copy-
paste the code from Slack into a new MATLAB m-file (the code will not run properly).
Double-click on the file, in the folder you just made, to open MATLAB.
Run the file by clicking the green "play" buUon in the Editor tab within MATLAB. Some calculaGons should appear in the
Command Window, and MATLAB should open a Figure window with a plot of the vector c as a point in space. Examine the
code in the m-file to idenGfy what commands caused the text to appear and the Figure window to appear.
Publish the code by clicking the "Publish as PDF" buUon on the Publish tab within MATLAB. There are more detailed
instrucGons on "publishing" m-files in a guide that is distributed with this homework assignment.
Combine the published PDF with a PDF of your answers to Problem 1, Problem 2, and Problem 3, and upload the single
combined PDF to Gradescope. (There are many ways to combine PDFs. We suggest using Adobe Acrobat, since Adobe CreaGve
Suite is free to all students at Boston University.)