Fall 2022 32A-4 Final (Solutions)
Full Nam e:
UIN:
Circle the nam e of your TA and the day of your discussion:
Gurkiran Dhaliwal Benjamin Major Advika Rajapakse
Tuesday Thursday
Instrucitons
• Read each Problem carefully.
• To get credit for a Problem, you must show all of your reasoning and calculations.
• Box your final answer.
• No calculators are allowed.
• You are allowed to use your formula sheet. The sheet should be hand-written, one-
sided, and of letter size.
• You may use the pages 2n —1 (2 ≤ n ≤ 10) for scratch work. Work found on those
pages will not be graded unless clearly indicated in the exam.
• Th er e w ill be 10 Pr oblem s in the exam . Problem 10 is True or False.
Problem Points Scores
1 10
2 10
3 15
4 15
5 10
6 15
7 15
8 15
9 20
10 10
Total 135
1
,Pr oblem 1 (10 points). Let v⃗, w⃗ be vectors on the xz-plane as described in the picture
⃗ = 6 and ||w⃗|| = 2 and the angle between them is equal to 4π .
below, where ||v||
1. (5 points) Compute the dot product v⃗ · w⃗.
2. (5 points) Compute the cross product v⃗ × w⃗ and express it in the coordinate form.
Solutions (Sketch). 1.
π √
v⃗ · w⃗ = ||v||||
⃗ w⃗|| cos( ) = 2.
4
6
2. π √
||v⃗ × w⃗|| = ||v||||
⃗ w⃗|| sin( ) = 6 2.
4
The right hand rule then yields:
√
v⃗ × w⃗ = ⟨0, −6 2, 0⟩.
2
, THIS PAGE IS INTENTIONALLY LEFT BLANK.
You may use this page for scratch work. Work found on this page will not be graded unless
clearly indicated in the exam.
3
Full Nam e:
UIN:
Circle the nam e of your TA and the day of your discussion:
Gurkiran Dhaliwal Benjamin Major Advika Rajapakse
Tuesday Thursday
Instrucitons
• Read each Problem carefully.
• To get credit for a Problem, you must show all of your reasoning and calculations.
• Box your final answer.
• No calculators are allowed.
• You are allowed to use your formula sheet. The sheet should be hand-written, one-
sided, and of letter size.
• You may use the pages 2n —1 (2 ≤ n ≤ 10) for scratch work. Work found on those
pages will not be graded unless clearly indicated in the exam.
• Th er e w ill be 10 Pr oblem s in the exam . Problem 10 is True or False.
Problem Points Scores
1 10
2 10
3 15
4 15
5 10
6 15
7 15
8 15
9 20
10 10
Total 135
1
,Pr oblem 1 (10 points). Let v⃗, w⃗ be vectors on the xz-plane as described in the picture
⃗ = 6 and ||w⃗|| = 2 and the angle between them is equal to 4π .
below, where ||v||
1. (5 points) Compute the dot product v⃗ · w⃗.
2. (5 points) Compute the cross product v⃗ × w⃗ and express it in the coordinate form.
Solutions (Sketch). 1.
π √
v⃗ · w⃗ = ||v||||
⃗ w⃗|| cos( ) = 2.
4
6
2. π √
||v⃗ × w⃗|| = ||v||||
⃗ w⃗|| sin( ) = 6 2.
4
The right hand rule then yields:
√
v⃗ × w⃗ = ⟨0, −6 2, 0⟩.
2
, THIS PAGE IS INTENTIONALLY LEFT BLANK.
You may use this page for scratch work. Work found on this page will not be graded unless
clearly indicated in the exam.
3
