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Day 1 – Geometric Vectors:Pgs. 127-128:6. Draw a vector to represent:the velocity of a fishing boat travelling at 8 knots (speed of one nautical mile per hour) on a heading of S75?W.the position of a city intersection 7 blocks east and 3 blocks south of your present position.the displacement of a crate that moves 6 m up a conveyor belt inclined at an angle of 18?.the force exerted by a chain hoist carrying a load of 200 kg.9. ABCD is a rhombus. For each of the following, find two vectors u and v in this diagram (expressed as point-to-point vectors) such that:u = v c) u = 2vu = -v d) u = 12v11. Determine the magnitude and the direction of each of the vectors in the given diagram. Express each direction as an angle measured counter-clockwise from a unit vector in the positive x direction.Pgs. 133-135:1. For each of the following, state a point-to-point vector equal to u+vb)a)and equal to u-v.3. Write a single point-to-point vector equivalent to each of these sums?a) PT+TS+SQ b) AC-GE+CE c) EA-CB+DB+AD d) PT-QT+SR-SQ4. Find the sum of the vectors u and v if θ is the angle between them.a) u=12, v=21, θ=70° b) u=3, v=10, θ=115° 5. A tour boat travels 25 km due east and then 15 km S50?E. Represent these displacements in a vector diagram, then calculate the resultant displacement.10. Simplify the following expressions using the properties of vector operations.c) 8(3x+5y)-4(6x-9y) d) 3x-6y+4(2y-x)-6x11. Let a=2i-3j+k, b=i+j+k and c=2i-3k. Find:b) a+2b-3cpgs. 141-1443. Find the resultant of each pair of forces acting on an object.a) forces of 7 N east and 12 N west c) forces of 6 N southwest and 8 N northwest4. Find the magnitude of the resultant of the four forces shown in the diagram.13. A man weighing 70 kg lies in a hammock whose ropes make angles of 20? and 25? with the horizontal. What is the tension in each rope?21. In order to keep a 250-kg crate from sliding down a ramp inclined at 25?, the force of friction that acts parallel to and up the ramp must have a magnitude of at least how many Newtons?24. Two tugs are towing a ship. The smaller tug is 10? off the port (left side) bow and the larger tug is 20? off the starboard (right side) bow. The larger tug pulls twice as hard as the smaller tug. In what direction will the ship move?pgs. 149-1501. A plane is heading due east. Will its ground speed be greater than or less than its airspeed, and will its flight path be north or south of east when the wind is from:a) N b) S 80? W c) S 30? E d) N 80? E2. A man can swim 2 km/h in still water. Find at what angle to the bank he must head if he wishes to swim directly across a river flowing at a speed of:a) 1 km/h b) 4 km/h3. A streetcar, a bus, and a taxi are travelling along a city street at speeds of 35,42, and 50 km/h, respectively. The streetcar and the taxi are travelling north; the bus is travelling south. Find:a) the velocity of the streetcar relative to the taxi b) the velocity of the streetcar relative to the bus7. A boat heads 15? west of north with a water speed of 3 m/s. Determine its velocity relative to the ground when there is a 2 m/s current from 40? east of north.8. A plane is steering east at a speed of 240 km/h. What is the ground speed of the plane if the wind is from the northwest at 65 km/h? What is the plane’s actual direction?Day 2 – Algebraic Vectors:pgs. 166-1692. Rewrite each of the following vectors in the form ai+bj.a) (5,2) b) (0,6) c) (1,6)3. Rewrite each of the following vectors as an ordered pair.a) 2i+j b) -3i c) 5i-5j4. Rewrite each of the following vectors in the form ai+bj+ck.a) (-2,1,1) b) (3,4,-3) c) (0, 4, -1) d) (-2,0,7) 5. Rewrite each of the following vectors as an ordered triple.a) 3i-8j+k b) -2i-2j-5k c) 2j+6k d) -4i+9j6. Express each of the following vectors as an algebraic vector in component form.a) u=12, θ=135° b) v=36, θ=330° 7. Express each of the following vectors as a geometric vector by stating its magnitude and direction.c) w=(4,3) d) x=(0,8) 8. What vector is represented in each of the following diagrams?b)c)a)e)f)d)12. For each of the following, draw the x-axis, y-axis, and z-axis and accurately draw the position vectors.a) OM=(6,-4,2) b) ON=(-3,5,3) 14. Find the magnitude of the following vectors.a) (-12, -4, 6) d) (-2, 23, 2) 17. a) Find the magnitude of the vector v=2i-3j-6k. b) Find a unit vector in the direction of v.20. Reposition each of the following vectors so that its initial point is at the origin, and determine its components.d)c)b)a)pgs. 172-1742. Find a single vector equivalent to each expression below.c) 0(4, -5) i) 2(0, 1, 0)-5(0, 0, 1) j) -124, -6, 8+32(4, -6, 8) 3. Simplify each of the following expressions.b) 3i-2j+3k-3-i+4j-3k 6. If a=3i+2j-k and b=-2i+j, calculate each magnitude.c) 2a-3b 8. Using vectors, demonstrate that these points are collinear.a) P(15, 10), Q(6, 4), and R(-12, -8) 16. a) Find the point on the y-axis that is equidistant from the points (2, -1, 1) and (0, 1, 3). b) Find a point not on the y-axis that is equidistant from the points (2, -1, 1) and (0, 1, 3).pgs. 178-1802. Calculate the dot product u?v given the magnitudes of the two vectors and the angle θ between them.a) u=3, v=4, θ=45° b) u=6, v=5, θ=60° 4. Find the dot product of each of the following pairs of vectors and state which pairs are perpendicular.a) a=(-1,3,4), b=(1,3,2) c) m=(-5,0,0), n=(0,-3,0) 5. a) Find three vectors perpendicular to (2, -3). b) How many unit vectors are perpendicular to a given vector in the xy-plane?6. a) Find three non-collinear vectors perpendicular to(2, -3,1). b) How many unit vectors are perpendicular to a given vector in three dimensions?8. Determine the angle between the following vectors.b) c=(5,6,-7), d=(-2,3,1) c) i=(1,0,0), m=(1,1,1)9. Given a=(2,3,7) and b=(-4,y,-14), a) for what value of y are the vectors collinear? b) for what value of y are the vectors perpendicular?pgs. 185-1862. Find u × v for each of the following pairs of vectors. State whether u × v is directed into or out of the page.a)b)3. State whether the following expressions are vectors, scalars, or meaningless.a) a?b×c b) a?b×b?c c) a+b?c d) a×b?c e) a×b?b×c f) a+b×c 4. Use the cross product to find a vector perpendicular to each of the following pairs of vectors. Check your answer using the dot product.a) 4,0,0 and (0,0,4) c) 2,-1,3 and (1,4,-2) 5. Find a unit vector perpendicular to a=4,-3,1 and b=2,3,-1.7. Express the unit vectors i, j, and k as ordered triples and show thata) i×j=k c) k×j=-i10. Given a=(2, 1, 0), b=(-1, 0, 3), and c=(4, -1, 1), calculate the following triple scalar/triple vector products.a) a×b?c Day 3 – Algebraic Vectors (Part 2):pgs. 192-1931. For each of the following, find the projection of u onto v and calculate its magnitude.a) u=(2,5), v=(6,4) c) u=(3,6,-2), v=(-4,3,8) 6. Calculate the area of the parallelogram with sides defined by the vectorsa) a=(1,2,-2) and b=(-1,3,0) b) c=(-6,4,-12) and d=(9,-6,18) 7. Find the area of the triangle with the given vertices.a) (7, 3, 4), (1, 0, 6), and (4, 5, -2)8. Find the volume of the parallelepiped determined by the vectors a=(2,-5,-1), b=(4,0,1) and c=(3,-1,-1). 9. For each of the following, calculate the work done by a force F that causes a displacement d, if the angle between the force and the displacement is θ.a) F=200 N, d=15 m, θ=49° b) F=4.3 N, d=2.6 m, θ=85°12. A pedicab is pulled a distance of 300 m by a force of 110 N applied at an angle of 6? to the roadway. Calculate the work done.14. A 35-kg trunk is dragged 10 m up a ramp inclined at an angle of 12? to the horizontal by a force of 90 N applied at an angle of 20? to the ramp. At the top of the ramp, the trunk is dragged horizontally another 15 m by the same force. Find the total work done.15. For each of the following, find the work done by a force F that causes a displacement d.c) F=(800,600), d=(20,50) d) F=12i-5j-6k , d=-2i+8j-4kpgs. 207-2084. Write each of the following vectors as a linear combination of i and j.a) the vector p=(-4,5) b) the position vector of the point A(8, -3)5. a) Can every vector in the xy-plane be written as a linear combination of u=(1,4) and v=(-2,5)? Explain why. b) Write the vector (-567,-669) in terms of u and v.6. Can every vector in the xy-plane be written as a linear combination of u=(-4,-6) and v=(10,15)? Explain why.7. Are the following sets of vectors coplanar?a) (1, -1, 1), (0, 1, 1), (1, 0, 2) b) (1, 0, 1), (1, 1, 1), (1, 0, -1)pgs. 213-2151. Given that w=au+bv, what can be said about w?a) if u and v are linearly independent? b) if u and v are linearly dependent?3. If u and v are linearly independent vectors, find the values of s and t for each of the following equations.a) su+2tv=0 d) su+7v=5u-tv10. Determine whether the following sets of vectors form bases for two-dimensional space. If a set forms a basis, determine the coordinates of v=(8, 7) relative to this base.a) v1=(1,2), v2=(3,5) d) v1=(3,5), v2=(6,10) 11. Determine whether the following sets of vectors form bases for three-dimensional space. If a set forms a basis, determine the coordinates of v= (1, 2, 3) relative to the basis.a) v1=(-1,0,1), v2=(2,1,1), v3=(3,1,1) ................
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