MATH 151 Fall 2014 Sections 549-551
Quiz #1 Solutions
Problem 1. If a circle has radius 10cm, what is the length of the arc subtended by a central angle of
72◦ ?
Solution. The formula for the length s of an arc subtended by a central angle of θ (in radians, not
degrees) in a circle with radius r is s = rθ. We are given that r = 10 cm, but we must convert 72◦ to
π
π
. Thus s = rθ = 10 · 72 180
= 4π cm. So the correct answer is (b).
radians by multiplying by 180
Problem 2. Find a unit vector that has the same direction as h3, −5i.
a
Solution. For a vector a = hx, yi, a unit vector in the same direction as a is given by |a|
, where |a| =
p
p
√
√
2
2
2
2
x + y is the ”length” of a. Here, we have a = h3, −5i, so that |a| = 3 + (−5) = 9 + 25 = 34.
Thus the correct answer is h √334 , √−5
i, corresponding to choice (a).
34
Problem 3. Express i and j in terms of a and b, where a = i − 2j and b = 3i + j.
Solution. One way to solve this problem is to first eliminate the term in a with j in it to get i in terms
of a and b. Substituting this value for i into the expression for b will then give j in terms of a and b.
Since j has a coefficient of −2 in the expression for a, and j has a coefficient of 1 in the expression for b,
we multiply b by 2 and add to a in order to eliminate j. This gives
a + 2b = (i − 2j) + (6i + 2j) = 7i,
which, after dividing both sides by 7 gives
i=
1
2
a + b.
7
7
Substituting equation (1) into the given expression for b gives
b=3
1
2 a + b + j,
7
7
or, after multiplying through and rearranging,
3
6
3
1
j = − a − b + b = − a + b.
7
7
7
7
Therefore, the correct answer is (b).
Problem 4. If a = h2, 3i and b = h−3, 8i, find 2a − 61 b.
Solution. We have 2a = h4, 6i and 16 b = h− 12 , 43 i. Thus
1
1 4
9 14
2a − b = h4, 6i − h− , i = h , i.
6
2 3
2 3
So, the correct answer is (b).
1
(1)