10.2,3,4. Vectors in 3D, Dot products and Cross Products

Transcription

1 Name: Section: 10.2,3,4. Vectors in 3D, Dot products and Cross Products 1. Sketch the plane parallel to the xy-plane through (2, 4, 2) 2. For the given vectors u and v, evaluate the following expressions. (a) 4u v (b) u + 3v u =< 2, 3, 0 >, v =< 1, 2, 1 > 3. Compute the dot product of the vectors and find the angle between them. Determine whether the angle is acute or obtuse. u =< 3, 2, 0 >, v =< 0, 0, 6 > 1

2 4. For the given vectors, find the orthogonal projection proj v u. u =< 13, 0, 26 >, v =< 4, 1, 3 > 5. Compute the cross product u v. u =< 0, 4, 0 >, v =< 0, 0, 8 > 6. Find a vector that is perpendicular to both u and v. u =< 0, 2, 2 >, v =< 0, 2, 2 > 2

3 Practice problems 1. The plane that passes through (2, 0, 0), (0, 3, 0), and (0, 0, 4). 2. For the given vectors u and v, evaluate the following expressions. (a) 4u v (b) u + 3v u =< 2, 1, 2 >, v =< 1, 1, 1 > 3. Compute the dot product of the vectors and find the angle between them. Determine whether the angle is acute or obtuse. u =< 10, 0, 4 >, v =< 1, 2, 3 > 3

4 4. For the given vectors, find the orthogonal projection proj v u. u =< 8, 0, 2 >, v =< 1, 3, 3 > 5. Compute the cross product u v. Find the are of the parallelogram that has two adjacent sides u and v. u = 3i j, v = 3j + 2k 6. Find the cross product u v and v u. u =< 4, 1, 1 >, v =< 0, 1, 1 > 4

5 Name: Section: Date: 1. The point (2, 3, 4) is Section 11.2 Quick Quiz Answer the following multiple choice questions by circling the correct response. (a) four units above the xy-plane (in the positive z-direction). (b) four units below the xy-plane (in the negative z-direction). (c) one unit from the origin. 2. The point ( 2, 0, 8) is (a) in the xy-plane. (b) in the yz-plane. (c) in the xz-plane. 3. The point ( 3, 4, 5) is (a) closer to the xy-plane than the xz-plane. (b) closer to the yz-plane than the xz-plane. (c) closer to the xz-plane than the yz-plane. 4. The vector with its tail at P( 5, 0, 3) and its head at Q( 6, 2, 8) is equal to the position vector (a) 11,2, 11. (b) 11, 2,11. (c) 1, 2, The magnitude of the vector joining P(5, 1, 5) to Q(4, 2, 6) is (a) 6. (b) 5. (c) A unit vector in the direction of v = 3, 4, 5 is (a) 3/5,4/5, 1. (b) 3/5,4/5, 1 / 2. (c) 3, 4, 5 / The plane x = 4 is parallel to the (a) xy-plane. (b) xz-plane. (c) yz-plane. 8. The set of points that satisfies the equation x 2 + 4x + y 2 + z 2 = 0 is (a) a sphere of radius 2 centered at ( 2, 0, 0). (b) the point (0, 0, 0). (c) a ball of radius 2 centered at ( 2, 0, 0). 9. The set of points that satisfies the equation x 2 2x + y 2 + 2y + z 2 2z is (a) a sphere of radius 1 centered at (1, 1, 1). (b) the point (1, -1, 1). (c) a ball of radius 2 centered at (1, 1, 1). 10. A vector parallel to 1, 2, 3 with magnitude 56 is (a) 1, 2, 3. (b) 2,4,6. (c) 4,2,6. Copyright 2015 Pearson Education, Inc.

6 Name: Section: Date: 1. If u v= 2.3, then Section 11.3 Quick Quiz Answer the following multiple choice questions by circling the correct response. (a) u is orthogonal to v. (b) u points away from v (π/2 < θ π). (c) u is roughly aligned with v (0 θ < π/2). 2. If u = 2,4, 1 and v = 4,3, 1 then u v equals (a) 21. (b) 19. (c) If u = 2, 4, 1 and v = 4,3, 4 then the angle between u and v is (a) acute (0 θ < π/2). (b) obtuse (π/2 < θ π). (c) a right angle. 4. The projection of u = 2i + 3j k onto i is (a) 2i. (b) 3i. (c) i. 5. The projection of u = 2,3,3 onto v = 3, 0, 4 is (a) 18 3, 0, (b) 18 3,0, 4 5. (c) 18 2,3, The scalar component of u = 2,3,3 onto v = 3, 0, 4 is (a) 18/25. (b) 18/5. (c) 5/ Given a vector u, the dot product uu equals (a) u. (b) the projection of u onto u. (c) u If a force F is applied to an object and the resulting displacement is the vector d, the work done by the force is (a) F d. (b) F d. (c) F d sin θ. 9. If u + v = u + v then (a) u and v are orthogonal. (c) u and v are unit vectors. (b) u and v are parallel and point in the same direction. 10. If v is a unit vector, the projection of u on v is (a) ( u v) v. (b) u v. (c) ( u v) u. Copyright 2015 Pearson Education, Inc.

7 Name: Section: Date: Section 11.4 Quick Quiz Answer the following multiple choice questions by circling the correct response. 1. Given two vectors u and v of fixed magnitude, the magnitude of their cross product is greatest when the angle between them is (a) π/2. (b) 0. (c) π. 2. Two vectors u and v are parallel if (a) u v = 1. (b) u v = 0. (c) u v = Two vectors u and v are orthogonal if (a) u v = 1. (b) u v = 0. (c) u v = The cross product k i equals (a) j. (b) j. (c) The cross product (2i k) ( i + 4j) equals (a) 4i j 8k. (b) 4i j 8k. (c) 4i + j + 8k. 6. The direction of (i + j ) (i j) is (a) k. (b) k. (c) i. 7. A 40-pound weight hangs from the end of a 2- foot bar that is attached to a wall at an angle of 45 o below the horizontal. The magnitude of the torque about the point on the wall where the bar is attached is (a) 40 2 ft-lbs. (b) 40/ 2 ft-lbs. (c) 20 ft-lbs. 8. The direction of the torque in Question 7 is (a) out of the page. (b) into the page. (c) in the plane of the page. 45 o 2 ft 40 lbs 9. The area of the triangle whose sides are u, v, and u + v is (a) u v. (b) u v /2. (c) u v. 10. If u and v are nonzero vectors, then u (u v) equals (a) u. (b) 1. (c) 0. Copyright 2015 Pearson Education, Inc.