Tute UV2 : VECTORS 1

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Aubrey Moody
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1 Tute UV2 : VECTORS 1 a b = ab cos θ a b = ab sin θ 1. A vector s is 4.2 m long and is directed at an angle of 132 anticlockwise relative to the x-axis as drawn below. Express s in i, j, k components. [ Answer: s = -2.81i j m ] 2. Given that a = 3i + 5j m and b = 5i + 2j m, find (i) s = a + b (ii) t = a b (iii) Express s and t in polar form (ie with magnitude and direction). [ Answer: The magnitude and direction of s are 7.28m and 106 relative to the +ve x axis. The magnitude and direction of t are 8.54m and 20.6 relative to the +ve x axis. ] 3. An aircraft flies on a northward course with an airspeed of 100 m s 1. It is carried slightly off course by a wind blowing from the west at 10 m s 1. Find the speed and direction of motion of the aircraft relative to the ground. [ Answer: The speed is 100.5ms 1 in the direction 5.7 east of north. ] 1 October 11,

2 4. Two forces act away from the same point. One force ( F 1 ) is 5 N in the x direction and the second ( F 2 ) is 10 N at an angle of 60 above the x axis. Find the magnitude and direction of the resultant force ( F). [ Answer: The resultant force is F=10i+5 3j N. The magnitude of the resultant force is 13.23N and is directed at an angle of 40.9 above the x axis. ] 5. A beam BP is hinged at a wall at point B and supported at the other end P by a cable as shown below: The tension F in the cable is 50N and the weight W of the beam is 60 N. Given that the condition for equilibrium of the beam is that the total force acting on it is zero, find the reaction R of the hinge on the beam both in Cartesian (i, j, k) and polar forms. [ Answer: R = 40i + 30j N where this reaction has a magnitude of 50 N acting at an angle of 36.9 above the beam. ] 6. A particle of mass m = 1 kg is acted upon by three separate forces. F 1, F 2 and F 3 act along OA, OC and OB respectively as shown. State the forces in Cartesian (i, j, k) form. Find the resultant acceleration a of the mass due to these forces. [ Answer: a = 40i + 30j N ] 2

3 7. What is the value of the dot product between two vectors a and b if the angle between them is 48 and a = 63 m and b = 4 m? [ Answer: m 2 ] 8. Given two vectors e = 3 i + 2 j + 4 k and f = 5 i + j + 3 k, find (i) e f (ii) the angle between e and f. [ Answer: e f = 29 θ = 24.5 ] 9. The work W done by a constant force F as it acts through a displacement s is W = F s What amount of work is done when a force F = 5 i 3 j + k N operates along the vector s = 3 i + j + 2 k m? [ Answer: W = 14 J ] 10. A carriage is caused to move along a straight railway line by a constant force F given by F = i 2j + 3k N. If the train moves from an initial point r i to a final point r f where r i = i + 7j m r f = 4i j m calculate the amount of work W done by this force. Find also the angle θ between these position vectors. [ Answer: W = 19 J θ = 95.9 ] 3

4 11. A rectangular box has sides a, b and c. Find an expression for the angle θ between the diagonals PQ and RS of any rectangular box. Hence determine this angle θ if the box is a cube. [ Answer: cos θ = a2 +b 2 +c 2 a 2 +b 2 +c 2 θ cube = 70.5 ] ] 12. Calculate the magnitude of the cross product of vectors a and b where a = 24 m, b = 48 m and the angle between them is 37. [ Answer: a b = 693 m 2 ] 13. The force F on a charge q that moves with velocity v in a magnetic field B is F = qv B. If a magnetic field of strength 2 Tesla is directed along the x axis and a particle with charge 10 6 C moves in the (x, y) plane at an angle of 45 to the +ve x direction with a velocity of 10 m s 1, find the magnitude and direction of the force experienced by this particle. [ Answer: The force has magnitude N, directed in the k direction. ] THE RIGHT HAND RULE If c = a b, then the direction of c is given by the right hand rule. It states that if you point the fingers of the right hand in the direction of the first vector 4

5 (a) and then curl your fingers towards the direction of the second vector (b), the thumb points to the direction of the cross product (c). For example i j = k j k = i i k = j NOTE : If c = a b then c is ALWAYS perpendicular to BOTH a and b. As the direction of the cross product can be found by curling the fingers of the right hand, the cross product is usually referred to as a curl, that is c equals a curl b. 14. The following two vectors a = 3i + j k and b = 2i + 3j + 4k define a plane. Find the unit vector û perpendicular to this plane. [ Answer: û = i j k or û = i j k ] 15. Show that the area A of a parallelogram with sides described by vectors a and b is given by A = a b If a = 2 i + j k m and b = i 3 j + k m define a parallelogram, find the magnitude of the area of this parallelogram. [ Answer: A=7.87 m 2 ] 16. Given two vectors f = 3 i + 5 j + 2 k g = 2 i 2 j 7 k determine the following: 5

6 (a) f g (b) the angle between f and g (c) the angle between f and (f g). [ Answer: f g = -31 i+25 j-16 k θ = θ = 90 ] 17. Given two vectors m = 4 i 6 j + 8 k n = 5 i + 3 j 11 k determine the following: (a) m n (b) the angle between m and n. [ Answer: (i) 42 i + 84 j + 42 k (ii) ] 18. The force F that acts on a unit positive charge moving with a velocity v in a magnetic field B is given by F = v B. Find the force on this charge if it is moving with a velocity v = 2 i + 3 j k m s 1 through a magnetic field B = i + 4 j + 5 k T. [ Answer: 19 i 11 j + 5 k N ] 19. The torque τ produced by a force F when it acts at a point which is displaced an amount r from an axis of rotation, is given by the cross product Find the torque if the force is τ = r F 6

7 F = 2i 3j + 4k N and the displacement is r = i + 2j + k m [ Answer: τ = 11 i 2j 7k N m ] 20. A bee with a mass m = 1 g is busily flying around a beehive. When its position vector r relative to the centre of the beehive is its velocity v is measured to be r = 20 i + 5 j 10 k cm v = 30 i 10 j + 5 k cm s 1. If the angular momentum L of the bee around the centre of the beehive is given by L = r mv (a) find the angular momentum of the bee in i, j, k coordinates (b) calculate the angle between the vectors r and v. [ Answer: (a) L = 10 7 ( 75 i j 50 k) kg m 2 s 1 ; (b) θ = ] 7

8 Further Practice Problems: 1. Find the polar coordinates r and θ of a point P having Cartesian coordinates x = 3 and y = What is the distance between point P having Cartesian coordinates (2,5) and point Q having coordinates (5,9)? 3. What are the polar coordinates of point Q in the previous question? 4. What angle does the segment P Q make with the axis x in the previous question? 5. What is the distance from the origin to point Q in practice problem 2? A point P has the following coordinates in a cylindrical system: (3,30,5). Find the coordinates in a Cartesian 3-D system. 6. What is the distance from point P to the origin in practice Problem 6? 7. A point P in a horizontal plane has coordinates (-3.50,6.20) m. Find the polar coordinates of this point. 8. Find the magnitude of the following vectors: (3.0,-5.0,6.0) m and (20,45,-30) N. 9. Is it possible for a vector to have zero magnitude but nonzero components? Explain 10. Let A = 2.0 i j and B = 3.0 i j. Using the properties of unit vectors i, j, and k, find the components of the vector C = A B. 11. Calculate A B for the vectors in Problem 10. What can you say about the directions of A and B? 12. Calculate A C and B C for the vectors in Problem 10. What can you say about their directions? 13. For the vectors in Problem 10, calculate A + B and B - A. What are the magnitudes of these two new vectors? 14. What can you say about the directions of the vectors calculated in Problem 13? Calculate the magnitude of vector C in Problem 10. [ Answer: r = 6.71, θ = 63.4 ; P Q = 5; r = 10.3, θ = ; θ = ; OQ = 10.3; x = 2.598, y = 1.5, z = 5; OP = 5.83; r = 7.12, θ = ; 8.37 m and N; No, A 2 = A 2 x + A 2 y + A 2 z; C = -13 k; A B = 0; the vectors are perpendicular to each other.; Same as for Problem 11; A + B = 5.0 i j and B - A = 1.0 i j. Both magnitudes equal 5.1.; The directions are perpendicular; The magnitude is 13. ] 8

9 B A A = 10, B = 8, C = 6, D = 5 B A A = 10, B = 8, C = 6 30 C D C A = 115, B = 60, C = 100 A C A 40 B A = 97, B = 132, C = B 30 C 15. For each of the above, calculate the sum of the vectors in polar form: 16. For a = 4i 3j and b = 6i + 8j, find the magnitude and angle to the x-axis of a, b, a + b, a b and b a students each walk a distance of 3m before turning and walking a further 4m so that they end up at distances of (a) 7m, (b) 1m, (c) 5m and (d) 6m from the starting point.what angle did each turn through? How far apart are students (a) and (b)? 18. Find AB if A and B have position vectors OA = i + 2j and OB = 3i 5j 19. Find the unit vector in the direction of each of the following vectors (a) 3i + 4j (b) 3i j (c) i + j (d) 0.3i + 0.7j (e) 6i 4j (f) 3 cos θi + 3 sin θj 20. If v = 2i + 3j, find (a) ˆv (b) 3ˆv (c) the vector s in the direction of v for which s = 6. (d) the unit vector in the direction of 2v 21. If a = 3i + 5j and b = 5i + 2j, calculate s = a + b and t = a b, also find s, t and the angles made by s and t with the x-axis. 9

10 [ Answer: (a) (a) 5.27, 61.1 (b) 6.23, (c) 71, 38.2 (d) 226, 17.2 (b) a = 5, θ = 323, b = 10, 53, a + b = 5 5, 27, a b = 5 5, 260, b a = 5 5, 80 (c) (a) 0 (b) 180 (c) 90 (d) 63, 8 m (d) 2i 7j (e) (a).6i +.8j (b).95i.31j (c).71i +.71j (d).39i +.92j (e).83i.55j (f) cos θi + sin θj (f) (a).55i +.83j (b) 1.65i j (c) 3.3i + 5j (d).55i.83j (g) s = 2i + 7j, s = 53, θ = 106, t = 8i + 3j, t = 73, θ = 21 10

Vectors Part 2: Three Dimensions

Vectors Part 2: Three Dimensions Last modified: 23/02/2018 Links Vectors Recap Three Dimensions: Cartesian Form Three Dimensions: Standard Unit Vectors Three Dimensions: Polar Form Basic Vector Maths Three