Which of these statements are true? A) 1, 2 and 3 only C) 2, 4 and 5 only. B) 1, 2 and 5 only D) 1, 3 and 4 only

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Harold King
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1 Name : 1 Qadrilateral RSTU is a parallelogram and M is the point of intersection of its diagonals. S M T ntoine lists the following vector operation statements: R U 1) ST + SR MU ) UT + UR SM 3) RS + RU RT 4) MT + MR + MS + MU 0 5) SR ST RT Which of these statements are tre? ) 1, and 3 only ), 4 and 5 only ) 1, and 5 only ) 1, 3 and 4 only Given the following information: a and b are nonzero vectors in the plane a b k is a scalar not eqal to zero k 1 Which of the following statements is tre? ) k( a b) ka kb ) k( a + b) ka + kb ) If a b 0 then a and b are collinear ) If a kb then a and b are noncollinear 3 Given and v two vectors that are not opposite. Which of the following is FLS? ) v v ) ( + v) + v ) 3v 6v ) + 3v 3 + v

2 4 Given the following prism having a rectanglar base. F H G Which vector is eqivalent to the resltant of the expression + H +? ) H ) F ) ) 5 The gyptians sed an ingenios plley system to move the blocks of stone sed in the constrction of pyramids. To minimize the work needed to displace the blocks, they applied a force oriented at 6. (Work (Nm) is the scalar prodct of the force vector and the displacement vector.) 1500 N 6 00 m Ronded to the nearest Nm, what work is needed to displace a block of stone horizontally for a distance of 00 m, if the force applied to it is 1500 N oriented at 6 o? ) Nm ) Nm ) Nm ) Nm 6 The following figre represents a right prism. H G F Which of these statements is FLS? ) + GF 0 ) 0 ) F 0 ) H + HF + FG G 0

3 7 Given the three vectors, v, and w. v (-, -3) and w are represented in the artesian plane below: y w x Which of the following statements is TRU? ) v and - are opposite. ) and v are eqivalent. ) w and ( v + w ) are perpendiclar. ) and 3 v are collinear. 8 Given vector parallelogram PQRS. Q R P S Which of the following statements is FLS? ) PQ + QR PR ) RP SP RS ) PS + SR RP ) SQ + QR + RS O

4 9 onsider rectangle shown below. Which of the following statements is tre? ) + ) + ) ) 10 Given vectors (-3, 9), v (6, ), w (6, -18) and k 0. Which of the following statements is FLS? ) k + kv k( + v) ) k v k w k ( v + w) ) and w are collinear. ) and v are orthogonal. 11 Given that and v are vectors, which of the following is NOT a vector? ) + v ) v ) v ) ( + v)

5 1 Given a and b, two vectors illstrated below. b a Which one of the following diagrams illstrates the relation between a and b and r, the resltant vector? ) r b ) b r a a ) r ) b r b a a 13 n airplane flying ast at 150 km/h enconters a 50 km/h wind blowing in a 30 ast of North direction. What will be the airplane's resltant velocity? ) 180 km/h [ 14 N] ) 00 km/h [N 30 ] ) 195 km/h [ 7 N] ) 13 km/h [ 19 S] 14 Given (3, ), and v (1, - 4) What are the components of the resltant of the following vector operation? v ) (1, 10) ) (, 6) ) (1, -6) ) (5, -6)

6 15 Given vectors and v shown below. v Which of the following vectors represents the resltant, r, of v? ) r ) r ) r ) r 16 Given vectors,,, below: Which proposition is TRU? ) + + ) + - ( + ) ) + + ) Vector (, -5) makes an angle of 40 with vector v whose magnitde is 7.8 nits. To the nearest tenth, what is the scalar prodct (dot prodct) of and v? ) 7.0 nits ) 3. nits ) 7.4 nits ) 4.0 nits

7 18 Given vectors and v. where (-, 3) and (6, 7) v (4, - 4) What is the scalar prodct of vectors and v? 19 Vectors and v are represented in the artesian plane below. where (3, 4) and (8, 14) v where (8, 1) and (5, -5) y v x What is the scalar prodct of vectors and v? 0 Given ( 9, ) and v ( a, b), two vectors that form a basis. Vector w (4, 16) can be expressed by the following linear combination: w + 3v. What are the components of vector v? 1 onsider the two vectors and v. The magnitde of is 10 cm at an angle of 140. The magnitde of v is 15 cm at an angle of 40. c + 3v What is the magnitde of c? Given vectors and v where: with (-5, 7) and (3, -5) v (6, 3) Find + v. Rond yor answer to the nearest tenth.

8 3 The scalar prodct of vectors d and f is 138. Their respective magnitdes are 7 and 5 nits. What is the measre of angle θ between vectors d and f? Rond yor answer to the nearest degree. d θ f 4 Given: (-1, 1) and ( 1, ) v. What are the components of ( 3v)? 5 On a compter screen, an alien ship was travelling at a very rapid speed. When it reached point (3, -), it sddenly exploded with one piece moving to point (-1, 3) and the other to point (5, 1). N (-1, 3) W v (5, 1) S (3, -) What is the sm of vectors and v? Give the magnitde of the resltant vector to the nearest nit, and its direction to the nearest degree. 6 Two nit vectors, and v, form a 60 angle as shown. What is the magnitde of the vector w if w + 3v? v 60

9 7 plane goes from city to city. In a artesian plane, city is at the origin and city has coordinates (100, 150). If there is no wind, the flight lasts one hor. Unfortnately, there is a wind. If the pilot does not adjst his flight path, he will be at point (10, 160) after an hor. What is the speed of the wind? 8 In qadrilateral illstrated below, points M, N, O and P are the midpoints of segments,, and respectively. P M O N Using the above figre, prove the following proposition: "The midpoints of the sides of any qadrilateral form the vertices of a parallelogram." 9 Given and v represented in the artesian plane below. y v 1 1 x What is the measre of the angle between these vectors, ronded to the nearest hndredth?

10 30 Peter and Marie are plling on an object. The forces they applied are 100 N and 50 N respectively bt in different directions: 40 and 10. The sitation is represented below. 100 N 50 N Tim is going to replace them. What force mst Tim apply to prodce the same effect on the object (strength and direction)? 31 Given the reglar hexagon on the right where a and b. F Prove the following identity: F a. 3 In the polygon below, G is a sqare. and G are the midpoints of sides G and F, respectively. Side is parallel to side F. G F Using the properties of vectors, show that + F + GF G.

11 33 Given the adjacent rhombs. Use vectors to prove the following statement: «The diagonals of the rhombs are perpendiclars.» 34 hot air balloon is flying de soth at 50 km/h. N Sddenly, the wind starts blowing from the sotheast at 8 km/h. N-W N- What is the reslting speed and direction of the hot air balloon? W S-W S- S 35 n airplane leaves airport and mst fly to airport. In the artesian plane on the right, these airports are represented by points and respectively. The scale of the graph is in kilometres. y O N S (400, 00) ring the flight, the plane enconters a steady wind. This wind is represented by (150, 15) the vector v (0, -15). x The pilot steers the plane so as to negate the effect of the wind. To the nearest degree, at what angle relative to the east shold the pilot point the plane in order to reach airport?

12 36 Triangle, shown on the right, is isosceles. Using vectors, show that median M is eqal to ( ) 1 M +. M 37 n airplane is flying de north at 00 km/h. N Sddenly, the wind starts blowing from the northwest at 50 km/h. N-W N- What is the reslting speed and direction of the airplane? W S-W S- S

13 nswers The scalar prodct of vectors and v is 16. The scalar prodct of vectors and v is -75. The components of vector v are (, 4). c cm Ronded to the nearest tenth + v is To the nearest degree, the angle measre is 38.

14 4 5 ( 3v ) ( ( -1, 1 ) 3 ( 1, )) ((-, ) ( 3, 6) ) ((- 3) + ( 6) ) ((- 6) + ( 1) ) ( 6) 6 6( -1, 1) (- 6, 6) nswer: The components of ( 3v) are (-6, 6). To the nearest nit, the magnitde is 8 nits. To the nearest degree, the direction is W76 N or eqivalent. 6 raw the vector. Since the adjacent angles in a parallelogram are spplementary, Therefore, sing the osine Law w + 3v w ()( 3) cos v 60 w 10 w 13 w nswer The magnitde of the vector is 3.6 nits.

15 7 Let ( x, y) be the wind vector y ( 100, 150) + ( x, y) ( 10, 160) (km) ( x, y) ( 10, 160) x 10 and x y 160 and y withot wind with wind Therefore ( x, y) ( 0, 10) The speed of the wind: ( 0, 10) ity 100 x (km) nswer The wind speed is approximately.36 km/h ( + ) 1 ( + ) MN M + N PO P + O + Vectors MN and PO are therefore parallel and of eqal length. Qadrilateral MNOP is therefore a parallelogram. Given ( 5, ) and v (,3) Scalar prodct v ( 5, ) (,3) v v 16 Magnitde of the vectors v ngle between and v v v cosθ v cos θ v cos θ θ nswer The angle between these two vectors measres

16 30 Measre of angle m since two consective angle in a parallelogram are spplementary. θ Resltant force (strength) F res cos100 F res N ( )( ) 100 F res 50 N N irection of resltant force sin 100 sin θ sin θ θ 4.38 The direction is , so abot nswer Tim mst apply a force of N with a direction of Hypothesis: 1. F is a reglar hexagon. a b a b -a onclsion : F a F -b Proof Reasons 1. a) + b) a + b 1. a) hasles' relation b) y sbstittion. a) - b) - a. a) and are non-collinear vectors by definition of a reglar hexagon. b) y sbstittion 3. a) F - b) F - b 3. a) F and are non-collinear vectors by definition of a reglar polygon. b) y sbstittion F a + a + b + - a + - b a 4. y vector addition

17 F + GF G to be proved 1. G + F + G by sbstittion since G and GF. G + + F + G becase F is the vector opposite to F 3. G G by sbstittion since F 4. G + G according to hasles Relation G + G and + 5. G G according to hasles Relation G + G + hasles Relation + hasles Relation Scalar prodct + + ( ) ( ) ( ) ( ) + as - definition of a rhombs + by distribtivity definition of scalar prodct c c c length of one side of the rhombs 0 Since 0, Scalar prodct theorem Given the resltant vector Magnitde of the resltant vector cos irection 44.7 sin 45 8 sin θ sin θ θ nswer: The reslting speed of the hot air balloon is 44.7 km/h in a direction of 6.7. v θ d

18 35 omponents of vector ( , 00 15) (50, 75) omponents of the nknown vector v +?? v? (50, 75) (0, -15)? (50 0, )? (30, 90) irection of the nknown vector 90 tan θ 30 θ 1.37 v?? 90 km θ 30 km nswer: To the nearest degree, the pilot shold point the plane at an angle of 1 relative to the east in order to reach airport. 36 1) ccording to hasles Relation M + M ) ccording to hasles Relation M + M 3) Hypothesis 1 M (Isosceles Triangle) 4) Similarly -1 M 5) y adding (1) and (), we get M + + M + M 6) Sbstitting in (3) and (4), we get 1-1 M ) Simplifying (6), we get M + 8) Therefore, 1 M ( + )

19 37 N W a 50 c 00 b? c 00 km/h, North a 50 km/h, from Northwest b? In Δ b a + c ac cos () b cos (45) b km lso sin ( ) sin ( ) sin ( ) a b c sin ( ) sin ( 45) sin ( ) ( 45) -1 sin sin ( 0.100) as shown or N nswer: The reslting speed of the airplane is km/h in a direction of 1.1 or N 1.1.

Correction key. Example of an appropriate method. be the wind vector x = 120 and x = y = 160 and y = 10.

Correction key. Example of an appropriate method. be the wind vector x = 120 and x = y = 160 and y = 10. Correction key 1 D Example of an appropriate method /4 Let x, y be the wind vector (km) y 100, 150 x, y 10, 160 100 x, 150 y 10, 160 100 withot wind with wind 100 + x = 10 and x = 0 150 + y = 160 and y