Vector Geometry Final Exam Review

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Vector Geometry Final Exam Review Problem 1. Find the center and the radius for the sphere x + 4x 3 + y + z 4y 3 that the center and the radius of a sphere z 7 = 0.

1 Vector Geometry Final Exam Review Problem 1. Find the center and the radius for the sphere x + 4x 3 + y + z 4y 3 that the center and the radius of a sphere z 7 = 0. Note: Recall x + ax + y + by + z = d ( are a, b, c ) and d + a 4 + b 4 + c 4, respectively. Answer: ( /3, /3, 1/) and.85, respectively. Problem. Describe the motion of the parametic equations x = sin(5t) and y = 3 cos(5t) on the interval π/4 t 3π/4. Problem 3. What is the Cartesian equation of the parametric equations Answer: The motion starts at (1/, 3/ ) and goes counterclockwise. x = cos ( cos(t) ) and y = 5 sin 3( cos(t) )? Problem 4. Suppose a partile travles on the path given by Answer: x + y/3 =. 5/3 x = 5t 3t + 5, y = t 3 + 3t + 5t 60 at time t, and another particle travels on the path given by x = 4s + s + 1, y = s 3 5s + s 17 at time s. Do they collide, and, if so, at what point? Answer: Collide at (73, 10). Problem 5. What graph represents r = cos(3θ) for π/6 θ 3π/4? Answer: Problem 6. What Cartesian equation is equivalent to the polar equation r = 7 cos(θ) sin(θ) + 1? Answer: x + 7yx + y =.

Let z = (3 + 3i). Find the nth roots of z, where n = 3. Answer: The roots are Answer: (f). n o 4e iπ 3iπ 17iπ 1, 4e 4, 4e 1. Problem 9. Let x = 1,, 6 and y = 8, 0, 7.

2 Problem 7. The point in the complex plane above could be (a) a square toot of i (b) a fourth root of 1 (c) a cube root of 1 (d) a square root of 1 + i (e) a cube root of 8 (f) a square root of 1 i Problem 8. Let z = (3 + 3i). Find the nth roots of z, where n = 3. Answer: The roots are Answer: (f). n o 4e iπ 3iπ 17iπ 1, 4e 4, 4e 1. Problem 9. Let x = 1,, 6 and y = 8, 0, 7. Find scalar projection of x onto y and vector projection of x onto y. Answer: 34/ 113 and 7/113, 0, 38/113, respectively. Problem 10. Two forces, F 1 and F, with magnitudes 13 lbs. and 11 lbs., respectively, act on an object at a point P as shown below. The angle F 1 makes with the positive x-axis is 4π/9 and the angle F makes with the positive x-axis is 7π/9. Find the resultant force F = F 1 + F. Answer: F = 6.17, 19.9.

3 Problem 11. Find point B if a vector AB starts at point A = (1,, ), ends at B, has magnitude 9, and is parallel to vector 9, 6,. Answer: B = (8.36, 6.91, 3.64). Problem 1. Find the angle (in degrees) located at vertex A of the triangle ABC with vertices A = (3, 6, 6), B = ( 5, 9, 4), C = (0, 1, 5). «11 180cos Answer: Problem 13. Use the right hand rule to determine if the components of a b are positive, negative, or zero. π. Answer: The x-component is positive, the y-component is positive, and the z-component is negative. Problem 14. Suppose an object with center of mass at the origin has a moment of inertia I = 30 ft lb sec and is initially at rest. If a force F = 4, 4, lbs. is applied at the point r =, 5, 6 feet and a force F = 4, 4, lbs. is applied at point r =, 5, 6 feet, use the formulas τ = I α and ω = t α to find the angular speed ω in revolutions per second after t = 8 seconds. Answer: ω = π. Problem 15. Which of the following sets of parameterized equations describes the line which passes through P = (5, 6, 7) and Q = ( 9, 5, 5)? (a) x = 14 39t y = 1 6t z = 6t (b) x = 3 39t y = 4 6t z = 3 6t (c) x = 8 31t y = + t z = 4 4t (d) x = 14 5t y = 1 6t z = 7t 3

4 (e) x = 19 31t y = 7 + t z = 9 4t (f) x = 33 14t y = 8 t z = 11 t (g) x = 9 5t y = 5 6t z = 5 7t (h) x = 5 14t y = 6 t z = 7 t Answer: (f). Problem 16. Given line L 1 : x = 1 5t, y = 4t + 3, z = 7t 6, determine whether line L : x = 5s 9, y = 9 s, z = 3 s is parallel to, skew to, or intersects L 1. If intersects, find the point of intersection. Answer: L intersects L 1 at ( 4, 7, 1). Problem 17. Which of the following equations describes a plane through the points P = ( 6,, 3), Q = (5, 5, 3), and R = ( 1, 5, 3)? (a) 9x 1y 3z = 0 (b) 16x + 1y + 4z = 3 (c) 1y + 0z = 0 (d) 9x + 33y + 40z = 0 (e) 6x + y + 54z = 8 (f) 4x 3y 34z = 6 (g) 18x + 66y + 9z = 36 Problem 18. Given the trajectory r(t) = ( e 3t) î + ( cos(3t) ) ĵ + (sin 1 (t))ˆk, find the velocity v(t). Answer: v(t) = `3e 3t î + ` 3 sin(3t) ĵ + Answer: (g). 1 1 t ˆk. Problem 19. Given the velocity v(t) = ( sin(4t) ) î + ( e 4t) ĵ + ( t ) ˆk, find the trajectory r(t) if r(1) = 0î + 0ĵ + 0ˆk. cos(4) Answer: r(t) = 1 ««1 4 4 cos(4t) î + 4e e 4t t 3/ ĵ + «ˆk Problem 0. A particle that passes the point P = (479, 969, 8) at time t = 3 is moving with velocity v(t) = 10t 4, 0t 4, 9t. Find the parametric equations for its motion. Answer: x = t 5 7, y = 4t 5 3, z = 3t Problem 1. A projectile is fired at ground level with an initial speed of 46 meters per second at an angle of elevation of 44. When and how far away will the projectile strike the ground? 4

5 Problem. Find the x-coordinate of the point on the curve x = 4t, y = 7t, z = 4t Answer: At time t = seconds, range is meters. which is at a distance of 10 units from (16, 8, 16), as measured along the curve in the positive t direction. Answer: Problem 3. On planet Q the acceleration of gravity is not a constant. Rather, its magnitude is given by the formula g(t) = 7t + 4 in feet/sec. A projectile is fired from 17 feet above ground level with a speed of 50 ft/sec at an elevation angle of 0. How high will the projectile be when t = 3? Answer: 4 feet high. Problem 4. Suppose a fixed cannon is to fire a projectile at an enemy tank, which is moving toward the cannon at a speed of 0 mph. If the cannon is to fire at the moment the tank is miles from the cannon, and the muzzle speed of the projectile is 950 mph, what is a correct equation to determine the firing angle? (The acceleration of gravity in these units is miles/hour.) Answer: `11.4 cos(θ) sin(θ) = 1. Problem 5. Find the unit tangent vector ˆT for the trajectory r(t) = e 3t, e t at the point corresponding to t = 3. Answer: ˆT = 1,

2. Evaluate C. F d r if F = xyî + (x + y)ĵ and C is the curve y = x 2 from ( 1, 1) to (2, 4).

2. Evaluate C. F d r if F = xyî + (x + y)ĵ and C is the curve y = x 2 from ( 1, 1) to (2, 4). Exam 3 Study Guide Math 223 Section 12 Fall 2015 Instructor: Dr. Gilbert 1. Which of the following vector fields are conservative? If you determine that a vector field is conservative, find a valid potential