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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)