1 Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate j i. (a) j (b) k k i In problems 3 through 8, let u 3i j k, v i 3j k and w i k. 3. Calculate u v. (a) 7 (b) 3 8,,. Calculate u v. (a), 0, (b) 8,,, 0, 7 5. Calculate u v w. (a) 30 (b) 8i j k. Calculate proj v u. 7, 7, 7 (a) (b) 7 7,, 7. Calculate cos where is the angle between u and v (a) (b) Which of the following vectors is orthogonal to both u and v? (a) k (b) 8i j k v u w 7
2 5 Chapter Vectors and the Geometry of Space 9. Which of the following is an orthogonal pair of vectors? (a) i j, i k (b) i j k, i j k 5i j k, i j 3k 3i k, j k 0. Which of the following is a set of parametric equations for the line through the points, 0, 3 and, 3, 3? (a) t (b) t t y t y 3t y t 3 t 3 3 t y 3t 3 3t. Write an equation of the plane that contains the line given by and is perpendicular to the line given by 7 y 5 3 y 3 7. (a) 7 y 7 0 (b) 3y 8 0 3y 0 7 y 7 0. Which of the following is a sketch of the plane given by y 3? (a) 3 (b) 3 3 y y 3. Calculate the distance from the point,, 3 to the line given by t, y, and t. 5 (a) (b) 3 3 y y 5
3 Chapter Test Bank 57. Find the center and the radius of the sphere given by y y 7 0. (a) Center:,, 3 (b) Center:,, 3 Center:,, 3 Radius: Radius: 37 Radius: Center:,, 3 Radius: Identify the quadric surface given by y. (a) Elliptic cone (b) Elliptic paraboloid Hyperbolic paraboloid Hyperboloid of one sheet (e) Hyperboloid of two sheets. Identify the quadric surface sketched at the right. (a) Hyperboloid of two sheets (b) Elliptic paraboloid Hyperboloid of one sheet Elliptic cone y 7. Find the equation of the surface of revolution if the generating curve, y, is revolved about the y-ais. (a) y y 0 (b) 3 0 y 0 3 y 0 8. Epress the cylindrical point, in spherical coordinates., 3 (a) 5, (b) 5,, 3, 0 5 5,, Find an equation in spherical coordinates for the surface given by the rectangular equation y. (a) (b) r sec 5,, 0.93 cot csc tan csc sec 0. Find the work done in moving a particle from P3,, 0 to Q, 3, if the magnitude and direction of the force are given by v 5,,. (a) 5 (b) cot csc sec
4 58 Chapter Vectors and the Geometry of Space Test Form B Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q, 0,. (a) i j 7k (b) i j k 5 i 5 j 7 5 k 5 i 5 j 5 k. Calculate j k. (a) i (b) k i j In problems 3 through 8, let u i j k, v 3i j k and w j k. 3. Calculate u v. (a) 7, 7, 7 (b) 3,,. Calculate u v. (a) 7, 7, 7 (b) 3,, 5. Calculate u v w. (a) (b) 0 i 8j. Calculate the component of u in the direction of v. (a) (b) 3 7. Calculate cos where is the angle between u and v. 9 9 (a) (b) 39 9 i 3 j 3 k
5 Chapter Test Bank Which of the following vectors is orthogonal to both u and w? (a) j (b) w u v j k 9. Which of the following statements is true about the vectors u v i 3 j k? i 3 j k and (a) u and v are orthogonal. (b) u and v are parallel. u is a unit vector of v. The angle between u and v is. 0. Which of the following is a set of parametric equations for the line through the points 3,, 0 and, 3, 3? (a) 7t (b) 3t 3 t y 3 t y 3 t y 5t 3 3t 3 3t 3 t y 3t 3t. Find an equation of the plane that passes through the points,,, 3,, 3 and,, 0. (a) y (b) 7 y y y 5 0. If the equation of a cylinder is given by 0, then (a) The rulings are parallel to the -ais. (b) The graph is a parabola. The rulings are parallel to the -ais. The generating curve is a parabola. 3. Calculate the distance from the point,, 3 to the plane given by 3 y 0. (a) (b) 5. Find the center and radius of the sphere given by the equation y y 0. (a) Center:,, (b) Center:,, Center:,, Radius: Radius: Radius: 0 Center:,, Radius: 0
6 0 Chapter Vectors and the Geometry of Space 5. Identify the quadric surface given by y. (a) Elliptic cone (b) Elliptic paraboloid Hyperbolic paraboloid Hyperboloid of one sheet (e) Hyperboloid of two sheets. Identify the quadric surface sketched at the right. (a) Hyperboloid of two sheets Hyperboloid of one sheet (b) Elliptic paraboloid Elliptic cone y 7. Find the equation of revolution if the generating curve, y, is revolved about the -ais. (a) y y 0 (b) 3 0 y 0 3 y Epress the spherical point 3, in rectangular coordinates., (a) 3 (b) 33, 33, 3, 3, 33 3, 33, 3 3 3, 33, 3 9. Find an equation in cylindrical coordinates for the surface given by the rectangular equation y. (a) r (b) r sec cot csc cot csc sec 0. Find the work done in moving a particle from P0,, to Q3,, if the magnitude and direction of the force are given by v,, 3. (a) 5 (b) 8 3
7 Chapter Test Bank Test Form C Chapter Name Class Date Section. A vector v has initial point, 3 and terminal point,. Find the unit vector in the direction of v. 3 (a) (b) 5 i 5 i 5 j 5 j 5 i 3 5 j i j 5. Find a vector with magnitude 0 in the direction of v,. (a) (b) 0, 0,, 5, 5 3. Forces with magnitudes of 00 and 300 pounds act on a machine part at angles of 0 and 5, respectively, with the positive -ais. Find the direction of the resultant forces. (a) 7. (b) Find the standard equation for the sphere that has points, 3, 5 and,, as endpoints of a diameter. (a) y 38 (b) y 3 y 3 5 y 3 5. Determine which vector is parallel to the vector v, 3,. (a),, (b) 3,, 3, 9, 3. Find the unit vector in the direction of the vector v i j k. (a) (b) i j k 9 i 9 j 5 i 5 j 3 5 k 9 k 3 i 3 j 3 k, 3,
8 Chapter Vectors and the Geometry of Space 7. Find u v if u 0, v 5, and the angle between vectors u and v is (a) 3003 (b) Find the vector component of u orthogonal to v for u i j and v i j. (a) 3 (b) i 3 i j j i j i 5 j 9. Calculate the angle that vector v 3i 5j k makes with the positive y-ais. (a) 59.5 (b) v v v (a) v (b) v proj u v v. Calculate u v for u 3i j k and v i j 3k. (a) i j k (b) 3i 7j 5k i j k i j k. Find the area of the parallelogram having vectors v i j k and v 3i j k as adjacent sides. (a) 3 (b) Find an equation of the plane determined by the points,, 3,, 3,, and 0,,. (a) 8i j 3k 0 (b) y 0 y 5 y 3 7. Find the point of intersection of the line given by the parametric equations 3 t, y 7 8t, and t with the y-plane. (a) 0, 5, 7 (b) 0, 37 3, 3 3, 7, 7, 33, 7
9 Chapter Test Bank 3 5. Find a generating curve for the surface of revolution given by y 3. (a) y 3 (b) 3 y Both (a) and (b). Write the equation in standard form and identify the quadric surface given by 9y 3 0. (a) 3 9y, Cone (b) 9 y, Hyperbolic paraboloid 9 y, Hyperboloid of one sheet 9 y 9, Hyperbolic paraboloid 7. Epress the spherical coordinate point, in cylindrical coordinates., (a) 3 (b) 33, 3, 33, 3, 3 33,, , 3, 3 8. Find a rectangular equation for the graph represented by the spherical equation cos. (a) y (b) y y 0
10 Chapter Vectors and the Geometry of Space Test Form D Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q, 0,.. Calculate i k. In problems 3 through 8, let u i j k, v i j k and w 3i k. 3. Calculate u v.. Calculate u v. 5. Calculate u v w.. Calculate proj v u. 7. Calculate where is the angle between u and v. 8. Calculate a vector perpendicular to both u and w. 9. Let L : t, y 5t, 7t and L : t, y 5 t, t. Show that the vectors v and parallel to lines L and L, respectively, are orthogonal. v 0. Find parametric equations for the line through the point,, 3 and parallel to the line given by y 5.. Write an equation of the plane that is determined by the lines given by and y 3.. Sketch the plane given by 3 y. y 3. Find the numbers and y such that the point, y, lies on the line passing through the points, 5, 7 and 0, 3,.. Write the equation of the sphere, y y 8 9 0, in standard form and find the center and radius Identify and sketch the quadric surface given by y 0.
11 Chapter Test Bank 5. Sketch the surface represented by the equation y. 7. Find the equation of the surface of revolution if the generating curve, 3, is revolved about the -ais. 8. Convert the point 3, 3, 7 from rectangular to cylindrical coordinates. 9. Find an equation in rectangular coordinates for the spherical coordinate equation and identify the surface. 9 csc csc 0. Find the area of the parallelogram having vectors u i j k and v 5i k as adjacent sides.
12 Chapter Vectors and the Geometry of Space Test Form E Chapter Name Class Date Section. A vector v has initial point, and terminal point, 3. Find the unit vector in the direction of v.. Find a vector with magnitude 3 in the direction of the vector v,. 3. Three forces with magnitude of 50, 0, and 0 pounds act on an object at angles of 0, 30, and 90, respectively, with the positive -ais. Find the direction and magnitude of the resultant forces.. Find the standard equation for the sphere that has points, 5, and 3, 7, 3 as endpoints of a diameter. 5. Determine whether each vector is parallel to the vector v i j 5k. If it is, find c such that v cu. a. u 8i j 0k b. u 8i j 0k c. u 3i 3 d. u i j 5 j 5 k k. A vector v has initial point,, 3 and terminal point,,. a. Write v in the component form. b. Write v as a linear combination of the standard unit vectors. c. Find the magnitude of v. d. Find the unit vector in the direction of v. e. Find the unit vector in the direction opposite that of v. 7. Find u v if u 7, v, and the angle between u and v is. 8. Let u i j k and v 3i j k. (a) Find the projection of u onto v. (b) Find the vector component of u orthogonal to v. 9. Find direction cosines for the vector with initial point,, 3 and terminal point, 3, Determine a scalar, k, so that the vectors u 3i j and v i kj are orthogonal.
13 Chapter Test Bank 7. Let u i j k and v 3i j k. a. Find u v. b. Show that the vector u v is orthogonal to v.. Calculate the area of the triangle with vertices 3,,,,, 5, and,,. 3. Find parametric equations for the line perpendicular to the lines given by 3 and y passing through the point,, y Find the point of intersection of the line y 3 and the plane 3 3y Sketch the surface represented by y.. Identify each of the quadric surfaces: a. 3 3y 3 3y 0 b. 5 y c. y 7. Consider the surface whose rectangular equation is y. a. Find an equation for the surface in cylindrical coordinates. b. Find an equation for the surface in spherical coordinates. 8. Sketch the surface represented by the spherical equation csc.
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
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