Fall 2014 – MATH 151, Sections 549-551
Quiz #2 Solutions
Problem 1. Find the cosine of the angle between the two vectors h2, 5i and h−1, 3i.
Solution. We will need to use both definitions of the dot product. Namely if a = ha1 , a2 i and b = hb1 , b2 i
are two vectors and and the angle between a and b is θ, then a · b = |a||b| cos θ = a1 b1 + a2 b2 . Then we
see that
a·b
.
cos θ =
|a||b|
Now,
h2, 5i · h−1, 3i = 2(−1) + 5(3) = −2 + 15 = 13.
and
|h2, 5i| =
and
|h−1, 3i| =
p
22 + 52 =
√
4 + 25 =
√
29,
p
√
√
(−1)2 + 32 = 1 + 9 = 10.
Therefore, the cosine of the angle θ between h2, 5i and h−1, 3i is
cos θ = √
13
13
√ =√
.
29 10
290
So the correct answer is (c).
Problem 2. Let m = h1, 2i and n = h4, 3i. Find the vector projection of m onto n.
Solution. The vector projection of m onto n, denoted projn m, is given by the formula
projn m =
m·n
n.
|n|2
Using the algebraic definition of the dot product, we have
m · n = 1(4) + 2(3) = 4 + 6 = 10.
We also have
|n| =
p
√
√
42 + 32 = 16 + 9 = 25 = 5.
Plugging these values into our formula, we see that
projn m =
10
2
8 6
h4, 3i = h4, 3i = h , i.
2
5
5
5 5
So the correct answer is (a).
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Math 151 Fall 2014
Quiz #2 Solutions
2
Problem 3. Which of the following is a Cartesian equation of the parametric curve determined by the
equations x(t) = cos2 (t), y(t) = sin(t), 0 ≤ t ≤ 2π.
Solution. Recall the Pythagorean trigonometric identity cos2 t + sin2 t = 1. Replacing cos2 (t) with x and
sin(t) with y, we get the Cartesian equation x + y 2 = 1. The correct answer is (d).
Problem 4. A horizontal force of 12 lbs is acting on a box as it is pushed up a ramp that is 10 ft long
and inclined at an angle of 30◦ above the horizontal. Find the work done on the box.
Solution. Since work is given by F · d, where F and d are the force and direction vectors, respectively,
we see from the definition of the dot product that work is given by |F||d| cos θ. We are given in the
statement of the problem that |F| = 12 lbs, |d| = 10 ft and θ = 30◦ . Plugging this into the expression
for work, we have
√
√
work = (12)(10) cos(30◦ ) = 120( 32) = 60 3 ft.lbs.
Thus, the correct answer is (b).