18.02 First Midterm SOLUTION
Spring 2020
Printed Name: SOLUTION
TA:
Section time:
Please turn cell phones off completely and put them away. No books, notes, or
electronic devices are permitted during this exam.
We encourage you to show your work. If you explain your reasoning, and if you
make a miscalculation, you will still earn most of the points for the problem. If the
problem says to explain your reasoning then you need to do so to get full credit.
Do not write on the backs of the pages! There are a couple pages of extra paper
in the back, which you can use. If you do so, please indicate on the page of each
problem where to find your answer.
(Do not write below this line.)
1 out of 10
2 out of 10
3 out of 10
4 out of 20
5 out of 20
6 out of 20
7 out of 15
Total out of 105
1
, 1a. (5 points) Find a unit vector in the same direction as the vector (−2, 1).
Solution: p √
The magnitude of the vector (−2, 1) is (−2) 2+1 = 5. So the unit vector in
√ √
the same direction is (−2/ 5, 1/ 5).
1b. (5 points) Find a unit vector in the opposite direction of the vector (−2, 1).
Solution: √ √
Multiplying the vector above by −1, we obtain (2/ 5, −1/ 5) which is a unit
vector in the opposite direction.
2
Spring 2020
Printed Name: SOLUTION
TA:
Section time:
Please turn cell phones off completely and put them away. No books, notes, or
electronic devices are permitted during this exam.
We encourage you to show your work. If you explain your reasoning, and if you
make a miscalculation, you will still earn most of the points for the problem. If the
problem says to explain your reasoning then you need to do so to get full credit.
Do not write on the backs of the pages! There are a couple pages of extra paper
in the back, which you can use. If you do so, please indicate on the page of each
problem where to find your answer.
(Do not write below this line.)
1 out of 10
2 out of 10
3 out of 10
4 out of 20
5 out of 20
6 out of 20
7 out of 15
Total out of 105
1
, 1a. (5 points) Find a unit vector in the same direction as the vector (−2, 1).
Solution: p √
The magnitude of the vector (−2, 1) is (−2) 2+1 = 5. So the unit vector in
√ √
the same direction is (−2/ 5, 1/ 5).
1b. (5 points) Find a unit vector in the opposite direction of the vector (−2, 1).
Solution: √ √
Multiplying the vector above by −1, we obtain (2/ 5, −1/ 5) which is a unit
vector in the opposite direction.
2
