3) Suppose r = 6, -8, 1, v = 2, -9, 6, and w = -9, -1, -3. Find (r + v ) w. A) 76 B) -12, -72, -6 C) 72, -17, 21 D) -76

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1 Exam Review Name Find the arc length of the given curve. ) x = cos t, y = sin t, z = 9t, -π t π A) π 85 B) 70π C) 85 D) π 85 ) Solve the problem. ) If v = -5i + j, find v. A) 7 B) 7 C) 9 D) 9 ) 3) Suppose r = 6, -8,, v =, -9, 6, and w = -9, -, -3. Find (r + v ) w. A) 76 B) -, -7, -6 C) 7, -7, D) -76 3) 4) Let a = 3j + k and b = j + k. Find a b a b. 4) A) B) C) 6 6 D) Find the angle between the given vectors to the nearest tenth of a degree. 5)u = i + 7j, v = -i - j A) 3.7 B) 0.9 C) 6.6 D) 74. 5) Find the vector projv u. 6)v = 3i - j + 3k, u = i + 0j + k A) 58 5 i j k B) 5 9 i j k C) 58 5 i j k D) 87 9 i j k 6) Solve the problem. 7) State whether the vectors v = 3i + 3j and w = 4i + 3j are orthogonal. A) not orthogonal B) orthogonal 7) 8) Find x so that the vectors v = i - xj and w = -4i - 3j are orthogonal. 8) A) 4 3 B) C) 3 4 D) ) Find the work done by a force of 00 pounds acting in the direction -i + j in moving an object 75 feet from (0, 0) to (-75, 0). 9) A) 5,000.0 ft-lb B) ft-lb C) 3,46. ft-lb D) ft-lb

2 Write the equation for the plane. 0) The plane through the point P(-6, 6, -7) and normal to n = -3i - 6j + 4k. A) -3x - 6y + 4z = -46 B) 3x + 6y - 4z = 8 C)6x - 6y - 7z = 8 D) -6x + 6y + 7z = 8 0) ) The plane through the point P(6, 4, ) and parallel to the plane 7x + 3y + 6z = 64. A) 7x + 3y + 6z = 66 B) 3x + 6y + 7z = 66 C)6x + 4y + z = 66 D) 7x + 3y + 6z = -66 ) Calculate the requested distance. ) The distance from the point S(6, -, -8) to the plane x + y + z = -0 ) A) 4 B) 8 3 C) 8 9 D) 4 3 Solve the problem. 3) If u = i + j + k, v = 0i + j + 6k, and w = 4i + 3j + 0k, evaluate (u v) w. A) -48 B) -96 C)00 D) -5 3) 4) Let a = -, 9, -4, b = -3,, -9, and c = 5, 8, -6. Evaluate a (b + c). 4) A) -95, -3, -8 B) -75, -3, -8 C) -95, 7, -8 D) -75, 7, 8 5) If v = -5i - 4j + k and w = -4i - 5j - k, find a vector perpendicular to both v and w. A) 9i - 9j + 9k B) -7i + j - 37k C) -3i + 5j - 35k D) -9i + 9j - 9k 5) 6) If w = 4i - 4j - k, find a vector perpendicular to both w and j + k. A) -i + 4j - 4k B) -4i + 8j + 4k C)4i - 4j - 3k D) -3i - 4j + 4k 6) 7) Calculate the area of the parallelogram with.v = -, 0, and w = -5, -, 0 as the adjacent sides. A) 33 B) 33 C) 33 D) 5 7) 8) Calculate the volume of the parallelepiped with edges a =, 4,, b = -,, 6, and c = -,, 4. A) B) 5 C)5 D) 69 8) 9) Find the equation of the plane containing the points P = (3,, 3), Q = (3, 0, -3), and R = (0,, -). A) 0x - 8y - 3z + 3 = 0 B) - y + 6z + 9 = 0 C)0x + 8y - 3z - 39 = 0 D) 0x + 8y - 3z = 0 9)

3 0) Find the equation of the plane perpendicular to the plane with equation x - y + 5z = 5 and the plane with equation 3x + z = -7 and containing the point P = (-5, 0, -7). 0) A) -x + 7y + 3z - 6 = 0 B) -x + 7y + 3z + 6 = 0 C)x - y + 5z + 40 = 0 D) 3x + z + = 0 The position vector of a particle is r(t). Find the requested vector. ) The velocity at t = 4 for r(t) = (3t + 4t + 7)i - 4t3j + (8 - t)k A)v(4) = 8i - 9j - 8k B)v(4) = 0i - 9j - 8k C)v(4) = 8i + 9j + 8k D)v(4) = 6i - 64j - 4k ) ) The acceleration at t = π 4 for r(t) = (4 sin t)i - (5 cos t)j + (3 csc t)k ) A)a π 4 = -6i - k B)a π 4 = 0j + k C)a π 4 = 6i + k D)a π 4 = -6i + k 3) The acceleration at t = 0 for r(t) = ti + (8t3-0)j tk A)a(0) = i + 8 k B)a(0) = i - k C)a(0) = i - 8 k D)a(0) = i - 6 k 3) Find the length of the curve with the given vector equation. 4)r(t) = 4ti + 3 cos tj + 3 sin tk; -9 t 6 A) 75 B) -75 C)375 D) -5 4) 5)r(t) = (t sin t + cos t)i + (t cos t - sin t)j ; -4 t 5 A) 9 B) 8 C)8 D) 4 5) Find parametric equations for the line described below. 6) The line through the points P(-, -, ) and Q(-3, 3, -5) A) x = -t +, y = 4t +, z = -6t- B) x = t +, y = t - 4, z = t + 6 C)x = -t-, y = 4t -, z = -6t + 6) 7) The line through the point P(3, 4, -5) parallel to the vector 4i - 3j - 5k A) x = -4t+ 3, y = -3t + 4, z = 5t - 5 B) x = -4t - 3, y = 3t - 4, z = -5t + 5 C)x = 4t - 3, y = -3t - 4, z = -5t + 5 D) x = 4t + 3, y = -3t + 4, z = -5t - 5 7) 3

4 Solve the problem. 8) Find symmetric equations for the line through the points P(-, -,-3) and Q(, -5,-5). A) x + = y = z + 3 B) x - = y = z C) x + 3 = y - -4 = z D) x - 3 = y + -4 = z + 3-8) 9) Find the symmetric equations of the line through (3,, ) and perpendicular to the plane x + y - z =. A) x + 3 = y + = z + - C) x = y - - = z - B) x - 3 = y - = z - - D) x = y + - = z + 9) 30) Find the symmetric equations of the line of intersection of the planes x - y + z = 0 and x + y + 3z = 6. A) x - -5 = y - - = z 3 C) x - 5 = y - = ẕ 3 B) x + 5 = y + = ẕ 3 D) x + -5 = y + - = z 3 30) 3) Find the equation of the plane containing the line x = -3 + t, y = 7 - t, z = 4 - t and the point (-3, 7, 4). 3) A) x - 3y - 4z = -8 B) x + 3y + 4z = -8 C)x + 3y - 4z = -8 D) x - 3y + 4z = -8 3) Find the equation of the plane through the point P(-6, 5, 7) and perpendicular to the line x = + 3t, y = t, z = 6 - t. 3) A) 3x + 4y - z = 5 B) 3x + 4y - z = 4 C)3x + 4y + z = -5 D) 3x + 4y - z = -5 Find the parametric equations for the line tangent to the curve at the given point. 33)r(t) = - cos t, -7t, -3 sin t at r(0) A) x = -t, y = -7, z = -3 B) x =, y = -7t, z = 3t C)x = -t, y = -7t, z = -3 D) x = -, y = -7t, z = -3t 33) Find the unit tangent vector of the given curve. 34)r(t) = (4 - t)i + (t - 9)j + (0 + t)k A)T(t) = - 9 i + 9 j + 9 k B)T(t) = - 3 i + 3 j + 3 k 34) C)T(t) = 3 i - 3 j - 3 k D)T(t) = 9 i - 9 j - 9 k 4

5 Find the curvature κ for the given function. 35)r(t) = (3t sin t + 3 cos t)i + 3j + (3t cos t - 3 sin t)k A) κ = - B) κ = 3t 3t C)κ = 3t D) κ = 9t 35) 36)r(t) = -3i + (0 + t)j + (t + 4)k A) κ = t + B) κ = C)κ = (t + ) 3/ D) κ = - (t + ) 3/ (t + ) 3/ 36) 37)r(t) = (9t sin t + 9 cos t)i + 9j + (9t cos t - 9 sin t)k A) κ = B) κ = 9t 8t C)κ = - 9t D) κ = 9t 37) Find T, N, and B for the given space curve. 38)r(t) = sin 3 4 ti cos 3 tj + 4tk 38) 4 39)r(t) = 8 3 ( + t) 3/i ( - t) 3/j + tk 39) For the curve r(t), write the acceleration in the form att + ann. 40)r(t) = 3 sin 4t + 7 i + 3 cos 4t - 3j + 5tk A)a = 44T C)a = 44T + 44N B)a = T + 44N D)a = 44N 40) 4)r(t) = (t - 3)i + ( t - 9)j + 8k t A)a = T + N B)a = t + t + C)a = t t + T + t + N t T + t + N t + D)a = t t + T + t + N 4) 4)r(t) = (4t sin t + 4 cos t)i + (4t cos t - 4 sin t)j + 9k A)a = 4t N B)a = 4tN C)a = 4T + 4t N D)a = 4T + 4tN 4) Solve the problem. 43) At what times in the interval 0 t π are the velocity and the acceleration vectors of the motion r(t) 5i + 5 cos (t)j + sin (t)k orthogonal? 43) 5

6 Sketch the space curve. 44)r(t) = 3 sin t, t, cos t, 0 t 3π 44) 45)r(t) = -t, 5 - t, 4 + t, 0 t 4 45) 6

7 Match the equation with the surface it defines. 46) y 4 + z = 46) A) Figure 3 B) Figure C)Figure D) Figure 4 47) x + z = y 4 47) A) Figure 3 B) Figure 4 C)Figure D) Figure 7

8 48) f(x, y) = 4 - x - y A) B) 48) C) D) Identify the type of surface represented by the given equation. 49) x = -4z, no limit on y A) Cylinder B) Hyperboloid of two sheets C) Sphere D) Parabolic cylinder 49) Find the curvature κ and radius of curvature R for the curve at the given point. 50) y = cos x, π 4, 50) Identify the type of surface represented by the given equation. 5) x 7 + z = y 8 5) A) Elliptic paraboloid B) Elliptic cone C) Hyperbolic paraboloid D) Ellipsoid 8

9 Answer Key Testname: REVIEW - MTH ) D ) D 3) D 4) B 5) D 6) D 7) A 8) A 9) B 0) A ) A ) A 3) D 4) A 5) A 6) D 7) A 8) C 9) C 0) B ) A ) D 3) C 4) A 5) A 6) C 7) D 8) A 9) B 30) A 3) D 3) D 33) D 34) B 35) B 36) B 37) A 38)T = 3 5 (cos 0.75t)i (sin 0.75t)j k; N = (-sin 0.75t)i - (cos 0.75t)j; B = 4 5 (cos 0.75t)i (sin 0.75t)j k 39)T = 3 + ti tj + 3 k; N = - ti + + tj ; B = ti tj + 3 k 40) D 4) A 4) D 43) t = 0; t = π ; t = π 9

10 Answer Key Testname: REVIEW - MTH 44) 45) 46) A 47) B 48) A 49) D 50) κ = 5) A 3 3, R = 3 3 0