Question: 1 2 3 4 5 6 Total
Marks: 10 6 8 4 4 8 40
Score:
Name (print):
Signature:
UWO ID number:
THE UNIVERSITY OF WESTERN ONTARIO
DEPARTMENT OF MATHEMATICS
MATHEMATICS 1600A
DISTANCE STUDIES
MIDTERM TEST
June 18, 2025 6:00 PM–8:00 PM
INSTRUCTIONS:
1. Check that this exam is 9 pages long and report immediately if there are any pages
missing. The exam is printed double-sided; there are 6 questions.
2. All questions must be answered in the space provided. Indicate your answer clearly.
Should you need extra space, a blank page is provided at the end of the booklet.
3. Show all your of your work and explain your answers fully. Unjustified, irrelevant or
illegible answers will receive little or no credit.
4. Do not unstaple the exam booklet.
5. No aids are permitted. In particular, calculators, cell phones, ipods etc. are
not allowed and may be confiscated.
6. If not stated otherwise, all vectors and equations involve real numbers.
Good luck!
, Mathematics 1600A Midterm Test 18 June 2025
1. For each of the following statements, indicate T if it is always true and F if it can be
false. Give a one-sentence justification for your answer. You will not receive credit
for your answer unless your justification is correct.
(a) (2 marks) There exist two unit vectors u and v in R3 such that
u·v =2
False
Solution: Using the formula for the dot product u • v = ∥u∥∥v∥ cos(θ) one can
conclude that
|u • v| = ∥u∥∥v∥| cos(θ)| ≤ ∥u∥∥v∥.
Adding the fact that u and v are unit vectors ∥v∥ = ∥u∥ = 1, we have
|u • v| = ∥u∥∥v∥| cos(θ)| ≤ ∥u∥∥v∥ = 1.
This implies that the inner product of any two unit vectors can be a number less
than 1 and larger than 1 (considering the meaning of absolute values) and it can
not be equal to 1.
(b) (2 marks) For any three vectors u, v, and w in R3 ,
u × (v × w) = (u × v) × w.
False
Solution:
To show that the statement is false, it is enough to provide a counterexample.
Let u = e1 , v = e1 and w = e2 . Then the left hand side will become
u × (v × w) = e1 × (e1 × e2 ) = e1 × e3 = −e2
On the other hand the right hand side is
(u × v) × w = (e1 × e1 ) × e2 = 0 × e2 = 0.
Page 2 of 9
Marks: 10 6 8 4 4 8 40
Score:
Name (print):
Signature:
UWO ID number:
THE UNIVERSITY OF WESTERN ONTARIO
DEPARTMENT OF MATHEMATICS
MATHEMATICS 1600A
DISTANCE STUDIES
MIDTERM TEST
June 18, 2025 6:00 PM–8:00 PM
INSTRUCTIONS:
1. Check that this exam is 9 pages long and report immediately if there are any pages
missing. The exam is printed double-sided; there are 6 questions.
2. All questions must be answered in the space provided. Indicate your answer clearly.
Should you need extra space, a blank page is provided at the end of the booklet.
3. Show all your of your work and explain your answers fully. Unjustified, irrelevant or
illegible answers will receive little or no credit.
4. Do not unstaple the exam booklet.
5. No aids are permitted. In particular, calculators, cell phones, ipods etc. are
not allowed and may be confiscated.
6. If not stated otherwise, all vectors and equations involve real numbers.
Good luck!
, Mathematics 1600A Midterm Test 18 June 2025
1. For each of the following statements, indicate T if it is always true and F if it can be
false. Give a one-sentence justification for your answer. You will not receive credit
for your answer unless your justification is correct.
(a) (2 marks) There exist two unit vectors u and v in R3 such that
u·v =2
False
Solution: Using the formula for the dot product u • v = ∥u∥∥v∥ cos(θ) one can
conclude that
|u • v| = ∥u∥∥v∥| cos(θ)| ≤ ∥u∥∥v∥.
Adding the fact that u and v are unit vectors ∥v∥ = ∥u∥ = 1, we have
|u • v| = ∥u∥∥v∥| cos(θ)| ≤ ∥u∥∥v∥ = 1.
This implies that the inner product of any two unit vectors can be a number less
than 1 and larger than 1 (considering the meaning of absolute values) and it can
not be equal to 1.
(b) (2 marks) For any three vectors u, v, and w in R3 ,
u × (v × w) = (u × v) × w.
False
Solution:
To show that the statement is false, it is enough to provide a counterexample.
Let u = e1 , v = e1 and w = e2 . Then the left hand side will become
u × (v × w) = e1 × (e1 × e2 ) = e1 × e3 = −e2
On the other hand the right hand side is
(u × v) × w = (e1 × e1 ) × e2 = 0 × e2 = 0.
Page 2 of 9
