MTH 112 Harvard University Vectors and Dot Products Questions just answer all the questions on the page, on a word file, not handwritten if possible. 5 hou

MTH 112 Harvard University Vectors and Dot Products Questions just answer all the questions on the page, on a word file, not handwritten if possible. 5 hours time MTH 112 FWH 9 – Vectors and Dot Products
1. Given the vectors u = h−6, −5i and v = h−2, 7i, compute the following:
a. |u|
e. u + v
b. The direction of u.
f. u − v
c. |v|
g. 2u − 3v
d. The direction of v
h. −u + 3v
2. Find the vector with the given magnitude and direction (given in standard position) written
in component form.
a. Magnitude: 7, Direction: 210◦
b. Magnitude: 4, Direction 15◦
3. A person starts walking from home and walks 6 miles at 40◦ north of east, then 2 miles at
15◦ east of south, then 5 miles at 30◦ south of west. If they walked straight home, in what
direction would they need to walk and for how far?
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4. Find the dot product between the given vectors.
a. Magnitude 7, Direction π/4; Magnitude
3, Direction −2π/3
b. h−2, 6i; h8, −3i
5. Find the angle between the given vectors.
a. h2, 5i; h3, −1i
b. h−4, −3i; h−6, −3i
6. Given the vectors a = h−1, 5i and b = h−3, −2i, find the magnitude of the projection of a
onto b and then find the projection vector of a onto b.
7. Find the work done against gravity by pushing a 45 pound cart 22 feet up a ramp that is 7◦
above horizontal. Assume there is no friction, so the only force is 45 pounds downward due
to gravity.
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