Calc 3 Quiz 1

# Calc 3 Quiz 1 - P(1 1 1 Q(2 1 and R(3-1 4 and get PQ = i-j...

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Instructor: Dr. Yuli B. Rudyak MAC 3474, Calculus 3 with Analytic Geometry, Fall 2011 QUIZ 1, SEPTEMBER 7, 2011 Problem 1. (a) For which values of b are the vectors h- 6 ,b, 2 i and h b,b 2 ,b i orthogonal? Solution: We have h- 6 ,b, 2 i · h b,b 2 ,b i = b 3 - 4 b . The vectors are orthogonal iﬀ h- 6 ,b, 2 i·h b,b 2 ,b i = 0. Since h- 6 ,b, 2 i·h b,b 2 ,b i = b 3 - 4 b , we get the equation b 3 - 4 b = 0. The solutions are b = - 2 , 0 , 2. (b) Find the angle between the vectors a = h 6 , - 3 , 2 i and b = h 2 , - 1 , 2 i . Solution: We use the formula cos θ = a · b | a || b | and get cos θ = 19 / 21. Problem 2. Find the scalar and vector projection of the vector a = 2 i + 6 j onto the vector b = 3 i - 3 j . Solution: We have comp b a = a · b / | b | = - 2 2 and proj b a = ( a · b ) b / b · b = - 2 i + 2 j . Problem 3. Find a vector that is perpendicular to the plane that passes through the points (1 , 2 , 1) , (2 , 1 , - 3), and (0 , 1 , 5). Solution: Denoting the points P (1 , 2 , 1) ,Q (2 , 1 , - 3) ,R (0 , 1 , 5), we get PQ = i - j - 4 k and PR = - i - j + 4 k . The desired vector can be found as PQ × PR = - 8 i - 2 k . You can also take 4 i + k . Problem 4. Find the area of the triangle whose vertices are (1 , 1 , 1), (2 , 0 , 1), and (3 , - 1 , 4). Solution: Similarly to what we did above, consider the points

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Unformatted text preview: P (1 , 1 , 1), Q (2 , , 1) and R (3 ,-1 , 4) and get PQ = i-j and PR = 2 i-2 j + 3 k . The area of the triangle is equal to | PQ × PR | / 2 = 3 √ 2 / 2. Problem 5. Find the volume of the parallelepiped determined by the vectors a , b and c where a = i + j-k , b = i-j + k , c = 2 i-2 j + 3 k . Solution: The volume is equal to the absolute value of the triple product abc . Thus V = ± ± ± ± ± ± 1 1-1 1-1 1-1 1 1 ± ± ± ± ± ± = 4 , since the determinant is positive. 1 Problem 6. A wagon is pulled a distance of 100 m along the horizontal path by a constant force of 50 N. The handle of the wagen is held at an angle of 30 ◦ above the horizontal. How much work is done? Answer: W = F · S = 50 · 100 cos 30 ◦ = 5000 √ 3 / 2 Nm....
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Calc 3 Quiz 1 - P(1 1 1 Q(2 1 and R(3-1 4 and get PQ = i-j...

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