Arnie Pizer Rochester Problem Library Fall 2005 WeBWorK assignment Vectors0Introduction due 01/03/2008 at 02:00am EST.

Transcription

1 WeBWorK assignment Vectors0Introduction due 0/03/2008 at 02:00am EST.. ( pt) rochesterlibrary/setvectors0introduction/ur vc 0.pg If Tom Bombadil s house is 5 miles east of Hobbiton and 2 miles south, what is the straight line distance (omit units)? 2. ( pt) rochesterlibrary/setvectors0introduction/ur vc 0 2.pg If the distance from the town of Bree to Weathertop is 2 miles on a 45 degree upward slope, what is the elevation gain (omit units)? 3. ( pt) rochesterlibrary/setvectors0introduction/ur vc 0 3.pg Frodo and Sam are studying a topographic map of Mordor. Place the letter describing contour lines on a map to the left of the number describing a possible goal.. If Frodo and Sam want to go directly uphill, they should go: 2. If Frodo and Sam want to find a level route, they should look at: 3. If Frodo and Sam want to find Mount Doom, they should look for: 4. If Frodo and Sam want to find the River Anduin, they should look for: A. Concentric contour lines B. Parallel contour lines C. Perpendicular to the contour lines D. Single contour lines 4. ( pt) rochesterlibrary/setvectors0introduction/ur vc 0 4.pg The nine Ring Wraiths want to fly from Barad-Dur to Rivendell. Rivendell is directly north of Barad-Dur. The Dark Tower reports that the wind is coming from the west at 59 miles per hour. In order to travel in a straight line, the Ring Wraiths decide to head northwest. At what speed should they fly (omit units)? 5. ( pt) rochesterlibrary/setvectors0introduction/ur vc 0 5.pg As Aragorn views the Dark Lord in a crystal ball of radius, he realizes that: The surface area of the ball equals: The volume of the ball equals: 6. ( pt) rochesterlibrary/setvectors0introduction/ur vc 0 6.pg As Gandalf falls into the depths of Moria, he begins to spin. If he wishes to slow his rate of spinning, he should do which of the following (type the appropriate letter)? A. Spread his arms wide B. Hug Himself C. Think of Galadriel D. Think of Sauron E. Wiggle His Nose 7. ( pt) rochesterlibrary/setvectors0introduction/ur vc 0 7.pg The population of Elves in Lorien is constant. If five Elves per day cross outward over the boundary of Lorien, and none ever return, then we can conclude that Elves in Lorien are being: A. Persecuted B. Anti-social C. Attracted by the promise of a better life D. Born 8. ( pt) rochesterlibrary/setvectors0introduction/ur vc 0 8.pg In the land of Mordor, where the shadows lie, it is afternoon. In which direction do the shadows point? You may assume that the Earth s axis of rotation is perpendicular to its plane of revolution about the sun. Type N, S, W, or E as appropriate. Type C, D, or I depending on whether the lengths of the shadows are constant, decreasing, or increasing. 9. ( pt) rochesterlibrary/setvectors0introduction/ur vc 0 9.pg Two dwarves decide to bore a tunnel through the center of the earth, connecting the mines of Moria with its antipode. They each have identical drills. One dwarf begins drilling from Moria and the other dwarf begins drilling from the antipode. When they meet at the center of the earth, are their two drills turning in the same direction? Type Y if yes, N if no.

2 WeBWorK assignment Vectorsspace3D due 0/04/2008 at 02:00am EST.. ( pt) rochesterlibrary/setvectorsspace3d/ur VC.pg What is the distance from the point (0, 6, 6) to the xz-plane? Distance = 2. ( pt) rochesterlibrary/setvectorsspace3d/ur VC 2.pg What do the following equations represent in R 3? Match the two sets of letters: a. a vertical plane b. a horizontal plane c. a plane which is neither vertical nor horizontal A. 5x B. 8 C. 3 D ( pt) rochesterlibrary/setvectorsspace3d/ur VC 3.pg Find an equation of the sphere with center (4, -, 0) and radius 4. = 0 Note that you must move everything to the left hand side of the equation and that we desire the coefficients of the quadratic terms to be. 4. ( pt) rochesterlibrary/setvectorsspace3d/ur VC 4.pg Find an equation of the sphere that passes through the origin and whose center is (9, -, 4). = 0 Note that you must put everything on the left hand side of the equation and that we desire the coefficients of the quadratic terms to be. 5. ( pt) rochesterlibrary/setvectorsspace3d/ur VC 5.pg Find the center and radius of the sphere x 2 + 2x + y 2 4y + z Center: (,, ) Radius: 6. ( pt) rochesterlibrary/setvectorsspace3d/ur VC 6.pg Find the equation of a sphere if one of its diameters has endpoints: (-4, -9, -) and (6,, 9). = 0 Note that you must move everything to the left hand side of the equation and that we desire the coefficients of the quadratic terms to be. 7. ( pt) rochesterlibrary/setvectorsspace3d/ur VC 7.pg Find an equation of the largest sphere with center (9,, ) that is contained completely in the first octant. = 0 Note that you must move everything to the left hand side of the equation that we desire the coefficients of the quadratic terms to be.

3 WeBWorK assignment Vectors2DotProduct due 0/05/2008 at 02:00am EST.. ( pt) rochesterlibrary/setvectors2dotproduct/ur VC 8.pg 8. ( pt) rochesterlibrary/setvectors2dotproduct/ur VC 5.pg Let a = (0, -6, -3) and b = (9, -7, 0) be vectors. Compute the following vectors. A. a + b = (,, ) B. 2a= (,, ) C. a - b= (,, ) D. a = 2. ( pt) rochesterlibrary/setvectors2dotproduct/ur VC 9.pg A child walks due east on the deck of a ship at 4 miles per hour. The ship is moving north at a speed of 9 miles per hour. Find the speed and direction of the child relative to the surface of the water. Speed = mph The angle of the direction from the north = (radians) 3. ( pt) rochesterlibrary/setvectors2dotproduct/ur VC 0.pg A horizontal clothesline is tied between 2 poles, 6 meters apart. When a mass of 2 kilograms is tied to the middle of the clothesline, it sags a distance of 4 meters. What is the magnitude of the tension on the ends of the clothesline? Tension = N 4. ( pt) rochesterlibrary/setvectors2dotproduct/ur VC.pg Find a b if a = 5, b = 8, and the angle between a and b is π 2 radians. a b = 5. ( pt) rochesterlibrary/setvectors2dotproduct/ur VC 2.pg If a = (9, 6, 3) and b = (0, 8, -6), find a b =. 6. ( pt) rochesterlibrary/setvectors2dotproduct/ur VC 3.pg What is the angle in radians between the vectors a = (-9, -8, -) and b = (4, 8, 0)? Angle: (radians) 7. ( pt) rochesterlibrary/setvectors2dotproduct/ur VC 4.pg Find a unit vector in the same direction as a = (-0, 4, -0). (,, ) Let a = (7, -4, -) and b = (-6, 9, -) be vectors. Find the scalar, vector, and orthogonal projections of b onto a. Scalar Projection: Vector Projection: (,, ) Orthogonal Projection: (,, ) 9. ( pt) rochesterlibrary/setvectors2dotproduct/ur VC 6.pg A constant force F = i 2j + 3k moves an object along a straight line from point (8, 3, 9) to point (9, 3, -0). Find the work done if the distance is measured in meters and the magnitude of the force is measured in newtons. Work: Nm 0. ( pt) rochesterlibrary/setvectors2dotproduct/ur VC 7.pg A woman exerts a horizontal force of 2 pounds on a box as she pushes it up a ramp that is 3 feet long and inclined at an angle of 30 degrees above the horizontal. Find the work done on the box. Work: ft-lb. ( pt) rochesterlibrary/setvectors2dotproduct/ur VC 8.pg Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (-3, 3) and arrived in the Iron Hills at the point with coordinates (-2, 6). If he began walking in the direction of the vector v = 4i + j and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn. (, ) 2. ( pt) rochesterlibrary/setvectors2dotproduct/ur VC F.pg If Yoda says to Luke Skywalker, The Force be with you, then the dot product of the Force and Luke should be: A. negative B. zero C. any real number D. positive

4 2

5 WeBWorK assignment Vectors3CrossProduct due 0/06/2008 at 02:00am EST.. ( pt) rochesterlibrary/setvectors3crossproduct/ur vc 2.pg You are looking down at a map. A vector u with u = 9 points north and a vector v with v = 4 points northeast. The crossproduct u v points: A) south B) northwest C) up D) down Please enter the letter of the correct answer: The magnitude u v = 2. ( pt) rochesterlibrary/setvectors3crossproduct/ur vc 2 2.pg Let a = (8,, 3) and b = (0, 5, 2) be vectors. Compute the cross product a b. (,, ) 3. ( pt) rochesterlibrary/setvectors3crossproduct/ur vc 2 3.pg If a = i + j + 5k and b = i + j + 2k Compute the cross product a b. i + j + k 4. ( pt) rochesterlibrary/setvectors3crossproduct/ur vc 2 4.pg If a = i+j+k and b = i+5j+k, find a unit vector with positive first coordinate orthogonal to both a and b. i + j + k 5. ( pt) rochesterlibrary/setvectors3crossproduct/ur vc 2 5.pg Find the area of the parallelogram with vertices (4,3), (9, 5), (, 6), and (6, 8).

6 WeBWorK assignment Vectors4PlanesLines due 0/07/2008 at 02:00am EST.. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 6.pg Find a unit vector with positive first coordinate that is orthogonal to the plane through the points P = (-3, -5, -5), Q = (, -, -), and R = (, -, ). (,, ) 2. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 7.pg A bicycle pedal is pushed straight downwards by a foot with a 42 Newton force. The shaft of the pedal is 20 cm long. If the shaft is π/5 radians past horizontal, what is the magnitude of the torque about the point where the shaft is attached to the bicycle? Nm 3. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 8.pg Enter T or F depending on whether the statement is true or false. (You must enter T or F True and False will not work.). Two planes parallel to a line are parallel. 2. Two planes perpendicular to a third plane are parallel. 3. Two planes either intersect or are parallel. 4. Two lines parallel to a plane are parallel. 5. A plane and a line either intersect or are parallel. 6. Two lines parallel to a third line are parallel. 7. Two lines perpendicular to a plane are parallel 8. Two lines either intersect or are parallel. 9. Two lines perpendicular to a third line are parallel. 0. Two planes perpendicular to a line are parallel.. Two planes parallel to a third plane are parallel. 4. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 9.pg Find a vector equation for the line through the point P = (2, -2, -) and parallel to the vector v = (, 3, -2). Assume r(0) = 2i 2j k and that v is the velocity vector of the line. r(t) = i + j + k 5. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 9a.pg Find a vector equation for the line through the point P = (-,, -3) and parallel to the vector v = (5,, -2). Assume r(0) = i + j 3k and that v is the velocity vector of the line.. r(t) = i + j + k Rewrite this in terms of the parametric equations for the line. 6. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 0.pg Given a the vector equation r(t) = ( t)i + ( t)j + (4 + 5 t)k, rewrite this in terms of the parametric equations for the line. x(t) = y(t) = z(t) = 7. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2.pg Given a the vector equation r(t) = ( + 5t)i + ( 4 + t)j + (0 4t)k, rewrite this in terms of the symmetric equations for the line. (quotient involving x) (quotient involving y) = (quotient involving z) = 8. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 2.pg Consider the planes 2x + 2y + 4 and 2x (A) Find the unique point P on the y-axis which is on both planes. (,, ) (B) Find a unit vector u with positive first coordinate that is parallel to both planes. i + j + k (C) Use parts (A) and (B) to find a vector equation for the line of intersection of the two planes,r(t) = i + j + k 9. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 3.pg (A) Find the parametric equations for the line through the point P = (5, -2, -4) that is perpendicular to the plane x 4y+4. Use t as your variable, t = 0 should correspond to P, and the velocity vector of the line should be the same as the standard normal vector of the plane. (B) At what point Q does this line intersect the yz-plane? Q = (,, ) 0. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 4.pg Consider the two lines L : 2t, + 2t, 3t and L 2 : 8 + 4s, + 4s, 4 + 2s Find the point of intersection of the two lines. P = (,, ). ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 5.pg Find an equation of the plane through the point (-3, 4, 3) and perpendicular to the vector (-3, -2, ). Do this problem in the standard way or WebWork may not recognize a correct answer. x + y + 2. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 6.pg Find an equation of the plane through the point (-, -4, 2) and parallel to the plane 2x 3y + 0. Do this problem in the standard way or WebWork may not recognize a correct answer. x + y + 3. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 7.pg Find the point P where the line + t, 2t, -3t intersects the plane x + y P = (,, )

7 4. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 8.pg Find the angle in radians between the planes 2x + and 5y ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 9.pg Find the distance from the point (, 2, 3) to the line 0, 2 + t, 3 + t. 6. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 20.pg Find the distance from the point (-2, 0, 4) to the plane 2x + 3y ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 2.pg Match the surfaces with the appropriate descriptions.. 2x 2 + 3y 2 2. x 2 + y 2 = 5 3. y 2 2x x 2 + 2y 2 + 3z 2 = 6. x x + 3y A. circular cylinder B. horizontal plane C. hyperbolic paraboloid D. elliptic paraboloid E. nonhorizontal plane F. ellipsoid G. parabolic cylinder 8. ( pt) rochesterlibrary/setvectors4planeslines/ur vc 2 22.pg A million years ago, an alien species built a vertical tower on a horizontal plane. When they returned they discovered that the ground had tilted so that measurements of 3 points on the ground gave coordinates of (0, 0, 0), (, 3, 0), and (0, 2, ). By what angle does the tower now deviate from the vertical? radians. 2

8 WeBWorK assignment Vectors5Coordinates due 0/08/2008 at 02:00am EST.. ( pt) rochesterlibrary/setvectors5coordinates/urvc 3.pg What are the rectangular coordinates of the point whose cylindrical coordinates are (r = 0, θ = 0, 6)? 2. ( pt) rochesterlibrary/setvectors5coordinates/urvc 3 a.pg What are the rectangular coordinates of the point whose cylindrical coordinates are (r = 2, θ = 5π 3, 3)? 3. ( pt) rochesterlibrary/setvectors5coordinates/urvc 3 2.pg What are the cylindrical coordinates of the point whose rectangular coordinates are ( 2,, 4)? r = θ = 4. ( pt) rochesterlibrary/setvectors5coordinates/urvc 3 3.pg What are the cylindrical coordinates of the point whose rectangular coordinates are (, 5, 3)? r = θ = 5. ( pt) rochesterlibrary/setvectors5coordinates/urvc 3 4.pg What are the rectangular coordinates of the point whose spherical coordinates are (5, 0π 6, 3π 6 )? 6. ( pt) rochesterlibrary/setvectors5coordinates/urvc 3 5.pg What are the spherical coordinates of the point whose rectangular coordinates are (3, 4, 4)? ρ = θ = φ = 7. ( pt) rochesterlibrary/setvectors5coordinates/urvc 3 6.pg What are the cylindrical coordinates of the point whose spherical coordinates are (4, 2, 4π 6 )? r = θ = z= 8. ( pt) rochesterlibrary/setvectors5coordinates/urvc 3 7.pg Match the given equation with the verbal description of the surface: A. Elliptic or Circular Paraboloid B. Cone C. Circular Cylinder D. Sphere E. Half plane F. Plane. r 2 + z 2 = 6 2. θ = π 3 3. φ = π 3 4. r 2 5. ρ = 2cos(φ) 6. r = 4 7. ρcos(φ) = 4 8. r = 2cos(θ) 9. ρ = 4 9. ( pt) rochesterlibrary/setvectors5coordinates/urvc 3 F.pg If an astronomer is using polar coordinates, then which of the following is the most likely object of study? A. the whole universe B. a globular cluster C. a solar system D. a planet