Cumulative Review of Vectors

Transcription

1 Cumulative Review of Vectors 1. For the vectors a! 1, 1, and b! 1, 4, 1, determine the following: a. the angle between the two vectors! the scalar and vector projections of a! on the scalar and vector projections of b! b on a!. a. Determine the line of intersection between p 1 : and p : 5 0. Determine the angle between the two planes.. If! and! are unit vectors, and the angle between them is 60, determine the value of each of the following: a. 4. Epand and simplif each of the following, where i!, j! 01! and k!! #!! 0! #! 0! #! , represent the standard basis vectors in a. 1i! 1i! j! 4j! k! R 41i! : 5k! 4j! 5k!!! #! 1i! j!!! 1i k i # 1j k 5. Determine the angle that the vector a! 14,, makes with the positive -ais, -ais, and -ais. 6. If a! 11,,, b! 1 1, 1,, and c! 1, 4, 1, determine each of the following: a. a! b! the area of the parallelogram determined b and!! d. c! a! b!! #! a b 1b a 7. Determine the coordinates of the unit vector that is perpendicular to a! 11, 1, 1 and b! 1,,. 8. a. Determine vector and parametric equations for the line that contains A1,, 1 and B11,,. Verif that C14, 1, is on the line that contains A and B. 9. Show that the lines L 1 : r! 1, 0, 9 t1 1, 5,,, and L : 5 10 are parallel and distinct Determine vector and parametric equations for the line that passes through (0, 0, 4) and is parallel to the line with parametric equations 1, t, and t,. 11. Determine the value of c such that the plane with equation c 8 0 is parallel to the line with equation 1 1. CHAPTERS

2 P(1,, 4) Q P = 0 5 N N 1. Determine the intersection of the line 5 with the plane Sketch the following planes, and give two direction vectors for each. a If P11,, 4 is reflected in the plane with equation , determine the coordinates of its image point, P. (Note that the plane is the right bisector of the line joining P11,, 4 with its image.) 15. Determine the equation of the line that passes through the point and intersects the line r! A11, 0, 1,, 4 s11, 1,, s R, at a right angle. 16. a. Determine the equation of the plane that passes through the points A11,,, B1, 0, 0, and C11, 4, 0. Determine the distance from O10, 0, 0 to this plane. 17. Determine a Cartesian equation for each of the following planes: a. the plane through the point A1 1,, 5 with n! 1, 5, 4 the plane through the point K14, 1, and perpendicular to the line joining the points (, 1, 8) and 11,, 4 the plane through the point 1, 1, and perpendicular to the -ais d. the plane through the points 1, 1, and 11,, 1 and parallel to the -ais 18. An airplane heads due north with a velocit of 400 km> h and encounters a wind of 100 km> h from the northeast. Determine the resultant velocit of the airplane. 19. a. Determine a vector equation for the plane with Cartesian equation 6 0, and verif that our vector equation is correct. Using coordinate aes ou construct ourself, sketch this plane. 0. a. A line with equation r! 11, 0, s1, 1,, s R, intersects the plane at an angle of u degrees. Determine this angle to the nearest degree. Show that the planes p 1 : 1 0 and p : are perpendicular. Show that the planes p : 1 0 and p 4 : 0 are parallel but not coincident. 1. Two forces, 5 N and 40 N, have an angle of 60 between them. Determine the resultant and equilibrant of these two vectors. 558 CUMULATIVE REVIEW OF VECTORS

3 a b. You are given the vectors and as shown at the left. a. Sketch a! b! a! b!,. Sketch a! 1. If a! b!. 16,,, determine the following: a. the coordinates of a unit vector in the same direction as a! the coordinates of a unit vector in the opposite direction to a! 4. A parallelogram OBCD has one verte at O10, 0 and two of its remaining three vertices at B1 1, 7 and D19,. a. Determine a vector that is equivalent to each of the two diagonals. Determine the angle between these diagonals. Determine the angle between OB! and OD!. 5. Solve the following sstems of equations: a d. 6. State whether each of the following pairs of planes intersect. If the planes do intersect, determine the equation of their line of intersection. a Determine the angle between the line with smmetric equations, 4 and the plane 8. a. If and are unit vectors, and the angle between them is calculate 16a! b! b! 5. a! #!! 60, 1a b. Calculate the dot product of and if and the angle between! and! 4!!!! 0! 0, 0! 0 4, is CHAPTERS

4 9. A line that passes through the origin is perpendicular to a plane p and intersects the plane at 1 1,, 1. Determine an equation for this line and the cartesian equation of the plane. 0. The point P1 1, 0, 1 is reflected in the plane p: 0 and has P as its image. Determine the coordinates of the point P. 1. A river is km wide and flows at 4 km> h. A motorboat that has a speed of 10 km> h in still water heads out from one bank, which is perpendicular to the current. A marina lies directl across the river, on the opposite bank. a. How far downstream from the marina will the motorboat touch the other bank? How long will it take for the motorboat to reach the other bank?. a. Determine the equation of the line passing through A1, 1, and B16,, 4. Does the line ou found lie on the plane with equation ? Justif our answer.. A sailboat is acted upon b a water current and the wind. The velocit of the wind is 16 km> h from the west, and the velocit of the current is 1 km> h from the south. Find the resultant of these two velocities. 4. A crate has a mass of 400 kg and is sitting on an inclined plane that makes an angle of 0 with the level ground. Determine the components of the weight of the mass, perpendicular and parallel to the plane. (Assume that a 1 kg mass eerts a force of 9.8 N.) 5. State whether each of the following is true or false. Justif our answer. a. An two non-parallel lines in R must alwas intersect at a point. An two non-parallel planes in R must alwas intersect on a line. The line with equation will alwas intersect the plane with equation k, regardless of the value of k d. The lines 1 and are parallel Consider the lines L 1 :, and L : k. k a. Eplain wh these lines can never be parallel, regardless of the value of k. Determine the value of k that makes these two lines intersect at a single point, and find the actual point of intersection. 560 CUMULATIVE REVIEW OF VECTORS

5 8. a. 5 1 t, 7 t, 7 t, 17. a , 1 4, r! 10, 0, 1 t14,, 7,, 7 t, a, b 4, c 1 a 4, 7 4, 7 b a 5, 8, 4 b 1 t s 7, t, s, s, 9. a t, 1 t, t, 1 1 t, t, 1 t, 10. a. These three planes meet at the point 1 1, 5,. The planes do not intersect. Geometricall, the planes form a triangular prism. The planes meet in a line through the origin, with equation t, 7t, 5t, a. r! 4 0 m! 1, n! 1, 5 s1, 1, 0, s R 1, 1, 0 11,, 1 0 Since the line s direction vector is perpendicular to the normal of the plane and the point 1, 1, 5 lies on both the line and the plane, the line is in the plane. 1 1, 1, The point 1 1, 1, 5 is on the plane since it satisfies the equation of the plane. d a , 0, a. r! 1,, 0. 1,, 0 t11,, 1, 15. a about a. 1 1, 1 t, t, 1. a. r! a 45, 0, b t111,, 5, ;. r! a 7, 0, 15 b ; r! t111,, 5, 17, 0, 1 t111,, 5, ; 1 5t, All three lines of intersection found in part a. have direction vector 111,, 5, and so the are all parallel. Since no pair of normal vectors for these three planes is parallel, no pair of these planes is coincident. a 1 a 1 and, 1, 1 b, 1, 1 b, a 9 7, 4 7, 7 b 5. A 6. a. r! 5, B, C 4 1 1, 4, 6 t 1 5, 4,, 7. about.6 units Chapter 9 Test, p a. 1, 1, a. 1 or or 1.. a. a 1, 1, 1 b, a 1, 1, 1 b, a 1, 1, 1 b, a 1, 1, 1 b, a 1, 1, 1 b, a 1, 1, 1 b a 1,, b 4t 1 t 5, 5, t, 14, 0, 5 4. a. 11, 5, 4 The three planes intersect at the point 11, 5, a. t t, 4 1 t 4,, The three planes intersect at this line. 6. a. m 1, n 1, 1 t, t, Cumulative Review of Vectors, pp a. about scalar projection: 14, 1 vector projection: a 5 169, , b scalar projection: 14, vector projection: a 8 9, 14 9, 8 9 b. a. 8 4t, t, t, about a. 4. a. 7i! 19j! 14k! ais: about 4.0, -ais: about 111.8, -ais: about a. 1 7, 5, 1 1 4, 0, 6 about 8.66 square units d a 1 and a 1, 1, 1, 0b, 0b 8. a. vector equation: Answers ma var. r! 1,, 1 t1 1, 5,, ; parametric equation: t, 5t, 1 t, If the -coordinate of a point on the line is 4, then t 4, or t. At t, the point on the line is 1,, 1 1 1, 5, 14, 1,. Hence, C14, 1, is a point on the line. 9. The direction vector of the first line is 1 1, 5, and of the second line is 11, 5, 1 1, 5,. So the are collinear and hence parallel. The lines coincide if and onl if for an point on the first line and second line, the vector connecting the two points is a multiple of the direction vector for the lines. (, 0, 9) is a point on the first line and 1, 5, 10 is a point on the second line. 1, 0, 9 1, 5, , 5, 1 k1 1, 5, for k R. Hence, the lines are parallel and distinct. A n s w e r s 687

6 10. vector equation: r! 10, 0, 4 t10, 1, 1, ; parametric equation: 0, t, 4 t, (0,, 6) (0, 0, 6) 1. a, 1 6, 1 6 b 1. a. (0, 0, ) (0, 0, 0) (1, 1, 1) (0,, 0) (, 0, 0) (6, 0, 0) (0,, 0) Answers ma var. For eample, 10,, and 16, 0,. (,, ) (0, 0, 0) (,, 1) Answers ma var. For eample, 1,, and 1,, 1. Answers ma var. For eample, 10,, 6 and 11, 1, , 15. q! 10, 0 11, 0, t1 11, 7,, 16. a about 1.49 units 17. a d km h, N 1.1 W 19. a. r! > 10, 0, 6 s11, 0, t10, 1,, s,. To verif, find the Cartesian equation corresponding to the above vector equation and see if it is equivalent to the Cartesian equation given in the problem. A normal vector to this plane is the cross product of the two directional vectors. n! 11, 0, 10, 1, , 10 11, ,, 1 So the plane has the form D 0, for some constant D. To find D, we know that (0, 0, 6) is a point on the plane, so D 0. So, 6 D 0, or D 6. So, the Cartesian equation for the plane is 6 0. Since this is the same as the initial Cartesian equation, the vector equation for the plane is correct. 0. a. 16 The two planes are perpendicular if and onl if their normal vectors are also perpendicular. A normal vector for the first plane is 1,, 1 and a normal vector for the second plane is 14,, 17. The two vectors are perpendicular if and onl if their dot product is ero. 1,, 1# 14,, Hence, the normal vectors are perpendicular. Thus, the planes are perpendicular. The two planes are parallel if and onl if their normal vectors are also parallel. A normal vector for the first plane is 1,, and a normal vector for the second plane is 1,,. Since both normal vectors are the same, the planes are parallel. Since , the point 10, 1, 0 is on the second plane. Yet since , 10, 1, 0 is not on the first plane. Thus, the two planes are parallel but not coincident. 1. resultant: about N, 7.6 from the 5 N force toward the 40 N force, equilibrant: about N, 14.4 from the 5 N force awa from the 40 N force. a. b a a b 688 A n s w e r s

7 . a. a + 1 b a 6 7, 7, 7 b 4. a. OC! 18, 9, a a 6 7, 7, 7 b 1 b BD! 110, 5 about 74.9 about a. t, 1 t, 1, 11,, 1, t, t, d. 1 s t, t, s, s, 6. a. es; 0, 1 t, t, no es; t, t, t, a r! t1 1,, 1,, , 1, 0 1. a. 0.8 km 1 min. a. Answers ma var. r! 16,, 4 t14, 4, 1, The line found in part a will lie in the plane if and onl if both points A1, 1, and B16,, 4 lie in this plane. We verif this b substituting these points into the equation of the plane, and checking for consistenc. For A: For B: Since both points lie on the plane, so does the line found in part a.. 0 km>h at N 5.1 E 4. parallel: 1960 N, perpendicular: about 94.8 N 5. a. True; all non-parallel pairs of lines intersect in eactl one point in R. However, this is not the case for lines in R (skew lines provide a countereample). True; all non-parallel pairs of planes intersect in a line in R. True; the line has direction vector (1, 1, 1), which is not perpendicular to the normal vector 11,, to the plane k, k is an constant. Since these vectors are not perpendicular, the line is not parallel to the plane, and so the will intersect in eactl one point. d. False; a direction vector for the line 1 1 is (, 1, ). A direction vector for the line is 4 1 4,,, or (, 1, 1) (which is parallel to 1 4,,. Since (, 1, ) and (, 1, 1) are obviousl not parallel, these two lines are not parallel. 6. a. A direction vector for L 1 :, is (0,, 1), and a direction vector for 14 L : k is (1, 1, k). k But (0,, 1) is not a nonero scalar multiple of (1, 1, k) for an k, since the first component of (0,, 1) is 0. This means that the direction vectors for L 1 and L are never parallel, which means that these lines are never parallel for an k. 6; 1, 4, Calculus Appendi Implicit Differentiation, p The chain rule states that if is a composite function, then d d du To differentiate an d du d. equation implicitl, first differentiate both sides of the equation with respect to, using the chain rule for terms d involving, then solve for d.. a. 5 9 d. 16 e f. 5. a. d (0, 1) 5. a a and a 5, 5 b 5, 5 b a a ; a d d d 8 Q d 4 R one tangent one tangent A n s w e r s 689