e a = 12.4 i a = 13.5i h a = xi + yj 3 a Let r a = 25cos(20) i + 25sin(20) j b = 15cos(55) i + 15sin(55) j

Transcription

1 Vetors MC Qld-3 49 Chapter 3 Vetors Exerse 3A Revew of vetors a d e f e a x + y omponent: x a os(θ 6 os( os(9.4 omponent: y a sn(θ 6 sn(9 0. a.4 0. f a x + y omponent: x a os(θ 5 os( omponent: y a sn(θ 5 sn( a g a x + y omponent: x a os(θ 5 os( os(6 3.5 omponent: y a sn(θ 5 sn(6 6.6 a h a x + y omponent: x a os(θ 3 os(35.6 omponent: y a sn(θ 3 sn(35 a a Let r a + a 6 5 a x + y omponent: x a os( θ 0 os( omponent: y a sn(θ 0 sn( a d a x + y omponent: x a os(θ 9 os(0 4.5 omponent: y a sn(θ 9 sn(0 7.8 a a 5os(0 + 5sn( os(55 + 5sn( r a

2 MC Qld-3 50 Vetors r 38.3 ( (0.84 tan(θ θ 33 The resultant vetor has a magntude of 38.3 n a dreton of 33 north of east. Let r a + tan(θ θ 7. The resultant vetor has magntude 3.5 n a dreton of 7. south of east. d Let r a + a 5os(45 + 5sn( os(30 + 0sn( r a r os( r 4.4 sn( α sn( sn(α 7sn(4 4.4 sn(α 0.40 α 3.7 The resultant vetor has magntude 4.4 n a dreton of 3.7 north of east. e Let r a + r 34.7 ( (0.6 tan(θ The resultant vetor has a magntude of 34.7 n a dreton of 36.4 north of east. Let r a + a 5.3.os( sn(80 43.os(37 +.sn( r a a os(60 + sn( os( sn( r a r 0.4 (7.8 + (7.57 tan(θ θ 46.5 The resultant vetor has magntude 0.4 n a dreton 46.5 north of east. f Let r r 3.5 ( ( 9.6

3 Vetors MC Qld r os( r 4.8 sn( α sn( sn(α 9sn( α α + θ 90 θ α The resultant vetor has magntude 4.8 n a dreton.5 south of east. g r r os( r 68.7 sn( α sn( sn(α 4sn( α α + θ 90 θ α The resultant vetor has magntude 68.7 n a dreton 6.9 south of east. R os( R 436 N 5 a a 4 and a 4 and ( a + + and d a 4 and e a 4+ and ( Let R a 400os( sn( R R ( ( N 7 a Let the two vetors e a and. a a os( θ 8 0 os( Let the two vetors e a and. a a os( θ 8 0 os( Let the two vetors e a and. a a os( θ 0 8 os( os(90 0

4 MC Qld-3 5 Vetors 8 a a ( + ( a ( 3( a (3 + ( The par of vetors n 8( are perpendular eause ther dot produt s equal to zero. 0 a a 3 + +, + 3 a (3 + + ( a , a ( ( a +, a ( + ( d a +, + 3 a ( + ( e a 4 + +, 3 a ( ( The par of vetors n 0(a are perpendular eause ther dot produt s equal to zero. a + 3 and n + are perpendular so ( + 3 ( n + 0 n n n n.5 n and n + 8 are perpendular so ( n ( n n n n 6 0 n 6 n ±4 n + 3 and + are perpendular so ( n + 3 ( + 0 n n n n 3 d + + and 3 + n+ 3 are perpendular so ( + + (3 + n n n n n 6 3 a Let the 3 fores e, and 3 and the resultng fore e R R os(30 + 5sn( os( sn( os(0 + 0sn( R ( ( R ( N 0.36 tan(θ 4.33 θ 0.5 The resultng fore has magntude 4.3N n the dreton of 0.5 south of east. Let the 3 fores e, and 3 and the resultng fore e R. R It s evdent from the dagram that of the fores have equal omponents and equal and opposte omponents, so + 35os( os( sn( os(30 + 5sn( R R ( N 9.5 tan(θ θ 9.8 The resultng fore has magntude 38.5N n a dreton 9.8 south of east.

5 Vetors MC Qld-3 53 Exerse 3B Vetor produt a a a sn( θ n ˆ 9 7sn(40 nˆ 40.5nˆ 40.5 n a downward dreton (nto the page a 40.5 n an upward dreton (or out of the page a asn( θ nˆ 5 0sn(40 nˆ 3.nˆ 3. n an upward dreton (out of the page a 3. n a downward dreton (or nto the page a asn( θ nˆ 4 8sn(80 nˆ 0nˆ 0 a 0 a ( ( 3+ ( + ( ( + ( 3 + ( ( a ( 3 + ( (6 4 ( (4 9 5 ( ( ( 3 + ( ( (3 + 3 ( ( ( ( a OA + + and OB 3 A a ( ( Therefore A ( 3 + ( square unts. OA + 3 and OB + A a ( + 3 ( + ( + 3 ( Therefore A ( square unts and r + 3 Torque r (3 + ( + 3 (3 + ( (6 0 ( (

6 MC Qld-3 54 Vetors ( 6 0 ( ( So torque ( Nm 6 a Torque r r r sn( θ nˆ sn( nˆ 40sn(0 nˆ 37.6nˆ 37.6ˆ (out of the page So torque ˆ 37.6 ( Nm Torque r r r sn( θ n ˆ sn( nˆ 40sn(60 nˆ 34.6nˆ 34.6ˆ (out of the page So torque ( Nm 7 a ( ( ( 0 d ( ( 8 a a + and + Let û e the unt vetor perpendular to a and, so û a a a ( + ( So û ( + + ( + ( ( ( + a and + Let û e the unt vetor perpendular to a and, so û a a a ( So û ( + ( + ( ( + 8 ( + ( ( + 9 a a ( ( + [4 ( ] ( + [4 + 4 ( ] ( + (0 + 8 ( Show that the result of the vetor produt a ( les n the plane of and Let a ( n + m 6 n(4 + m( 6 4n + m m 6 (4n+ m m Therefore, 6 m 0 4n + m m 8 0 4n + ( 8 6 4n n 4 Therefore, a ( 4 8, so a ( must le n the plane of and. Show that, n general, the vetor a ( les n the plane of and. The vetor ( s perpendular to the plane of and, so f a vetor s n the plane of and, t must e perpendular to ( and ve versa. Therefore, sne the vetor a (, wll always e perpendular to the vetor (, t wll always le n the plane of and. 0 Tae B as the orgn wth the x-, y- and z-axes taen as usual. BA 3 + AC AC ( (

7 Vetors MC Qld The magntude of the vetor s. So 00 ( Torque r BA ( ( (3 + ( (3 + ( (0 6 ( ( So torque ( Nm Tae A as the orgn wth the x- and y-axes taen as usual and the postve z-axs as out of the page AB.5 40os( sn( Torque r AB (.5 ( (.5 ( (0 0 (0 0 + ( So torque Nm Exerse 3C Salar trple produt a a + +,, a a a3 a (8 + 3 (0 + ( a, +, a a a3 a (3 + ( a +, 3 +, + a a a3 a ( 4 ( d a + +,, + 3+ a a a3 a ( + 3 (0 + + ( e a 4 + +,, + +

8 MC Qld-3 56 Vetors a a a3 a ( + ( + + ( f a + +,, a a a3 a ( + 6 ( ( g a + 3+, +, + + a a a3 a ( + 3( + + ( a a +, 3, + Volume a a Volume a u unts a + 3, +, + 3 Volume a 3 a Volume a 9 9 u unts a 3+, + +, Volume a 3 a Volume a 0 0 u unts d a, + +, + 3 Volume a a Volume a 4 4 u unts 3 a a +, 3 + and n a 0 a 3 0 n n 4 n 3 (0 3 ( n ( n 9 n 4 So n 4 0 n 4 a + n+, + +, 3+ a 0 n a 3 n ( + 3 n(4 + ( 6 5 n 8 n 3 So n n n.5 a 3 +, 3 + n 3, + 4 a 0 3 a 3 n 3 4 n n ( 4n + 3 ( + 3 (3 n n n 4 n

9 Vetors MC Qld-3 57 So 4 n 0 n 4 4 a Let the ottom left-hand orner e the orgn wth the x-, y- and z-axes defned as usual. Let a 8 0sn(40 + 0os( Volume a 0 0 a ( Volume a u unts Let the ottom left orner e the orgn wth the x-, y- and z-axes defned as usual. Let a 9sn(44 + 9os( The dagram elow s the horzontal or xz-plane Let 4os(70 4sn( Volume a a 0 Volume a u unts a ( ( d e ( f ( 0 6 If a s a lowse orderng of,, and, the result s. If t s antlowse, t s. 7 a If a, and are oplanar, the volume of the parallelpped formed s zero. If a s n the same plane as and, then a s perpendular to. Therefore the salar produt wll e 0. Exerse 3D Applatons of the vetor produt a P (,,, Q (, 0, and R ( 3,, 0 PQ ( + (0 + ( QR ( 3 + ( 0 + (0 4 + n PQ QR ( 0 ( 0 + ( 4 6 n (x a + n (y + n 3 (z 0 (x (y 0 6(z 0 x y 6z x y 6z x y 6z 5 P(, 0, 0, Q(0,, 0 and R(0, 0, PQ (0 + ( 0 + (0 0 +

10 MC Qld-3 58 Vetors QR (0 + (0 0 + ( 0 + n PQ QR ( 0 ( 0 + ( n (x a + n (y + n 3 (z 0 (x + (y 0 + (z 0 0 x + y + z 0 x + y + z P(3,,, Q(,, and R( 3,, PQ ( 3 + ( + ( QR ( 3 + ( + ( 4 + n PQ QR ( 0 n (x a + n (y + n 3 (z 0 0(x 3 + 0(y (z 0 (z 0 z 0 z d P(,,, Q(4,, 0 and R( 3,, 3 PQ (4 + ( + (0 3 QR ( ( + ( n PQ QR ( 3+ 3 (9 7 + (9 7 + n (x a + n (y + n 3 (z 0 0(x (y + + (z 0 (y + + (z 0 y + z 0 y + z 0 y + z e P(, 0,, Q(3, 3, and R(3,, PQ (3 + (3 0 + ( + 3 QR (3 3 + ( 3 + ( + n PQ QR (3 0 ( 0 + ( 0 3 n (x a + n (y + n 3 (z 0 3(x (y 0 (z 0 3x 6 y z + 0 3x y z 5 0 3x y z 5 3x + y + z 5 f P(3,,, Q(,, and R(, 3, PQ ( 3 + ( + ( + QR ( + (3 + ( 3 + n PQ QR (0 ( ( 0 4 n (x a + n (y + n 3 (z 0 (x 3 4(y (z 0 x + 3 4y + 8 z + 0 x 4y z + 0 x 4y z x + 4y + z a n +, (,, n (x a + n (y + n 3 (z 0 (x + 0(y + (z 0 x + z 0 x + z 4 0 x + z 4 n + +, (0,, n (x a + n (y + n 3 (z 0 (x 0 + (y + (z 0 x + y + z 4 0 x + y + z 5 0 x + y + z 5 n +, (,, 0 n (x a + n (y + n 3 (z 0 (x + (y (z 0 0 x + y z 0 x + y z 4 0 x + y z 4 d n 3 +, (, 3, n (x a + n (y + n 3 (z 0 3(x (y 3 + (z 0 3x 3 y z 0 3x y + z 0 3x y + z 3 n, (0, 0, 0 n (x a + n (y + n 3 (z 0 0(x 0 + (y 0 (z 0 0 y z 0 4 a AB AO + OB OA + OB

11 Vetors MC Qld-3 59 OA 3 OB 5 So AB ( AC 5 n AB AC (0 0 + ( ( + 3 Any multple of n s normal to the ABC plane so + 3 s a normal. d n + 3, pont (0, 0, 0 n (x a + n (y + n 3 (z 0 0(x 0 + (y (z 0 0 y + 3z 0 5 Pont (,, 3 Plane perpendular to y-axs: n n (x a + n (y + n 3 (z 0 0(x + (y + 0(z 3 0 y 0 y 6 a At P y 0 and at Q x 0 x + 3y 6 y 0 x x 6 x 3 P (3, 0 x y 6 3y 6 y Q (0, Pont (0, 0, and ontans lne x + 3y 6 Let R (0, 0, rom part (a, P (3, 0, 0 and Q (0,, 0 PQ (0 3 + ( 0 + ( QR (0 0 + ( 0 + (0 n PQ QR ( 0 (3 0 + ( n (x a + n (y + n 3 (z 0 (x 0 3(y 0 6(z 0 x 3y 6z x 3y 6z 6 x + 3y + 6z 6 7 3x + 4y + z 6 n ˆn n n ( ( ( ( z 3x + y 3x + y z 0 n 3 + ˆn n n ( ( ( ( m v r 4 + L r mv (4 + (3 + 3 (4 + ( (4 0 a m 3 v + 3 P mv 3( r 4 L r P 4 ( ( a Show that v and r are perpendular r ros( θ + rsn( θ v vsn( θ + vos( θ r v ( os( sn( ( sn( os( r θ + r θ v θ + v θ ros( θ ( vsn( θ + rsn( θ vos( θ

12 MC Qld-3 60 Vetors rvos( θsn( θ + rvos( θsn( θ 0 Sne the dot produt s zero, r and v must e perpendular. show that L 0 L r mv ( ros( θ + rsn( θ m( vsn( θ + vos( θ ( ros( θ + rsn( θ ( mvsn( θ + mvos( θ ros( θ rsn( θ 0 mvsn( θ mvos( θ 0 r sn( θ 0 mv os( θ 0 r os( θ 0 mvsn( θ 0 ros( θ rsn( θ + mvsn( θ mvos( θ r ( os( θ mvos( θ + rsn( θ mvsn( θ mvr(os ( θ + sn ( θ mvr So L ( mvr mvr, as requred. v s halved d v s douled e As her arms move to her sde, r dereases whh auses a orrespondng nrease n v due to onservaton of angular momentum. Ths auses her to spn faster. a The stuaton desred s shown n the dagram elow It an e seen from the dagram that d wll gve the shortest dstane from the orgn to the plane. nop n OP os( α d But os(α OP d OP os( α So nop nd d nop n Therefore the shortest dstane etween the plane and the orgn s gven y nop n x + 3y z n + 3 P (,, 7 (any pont that satsfes the equaton ( + 3 ( + 7 Shortest dstane ( + 3 ( ( a If n s normal to the frst plane and n normal to the seond, the lne of nterseton (whh s on oth planes must e normal to oth n and n, The ross produt n n fnds a vetor that s normal to oth n and n, so t must e parallel to the lne of nterseton of the two planes. Therefore, n n gves the dreton of the lne of nterseton of the two planes. x 3y + z 0 and 3x y + z 3x y z 0 Let n 3+ and n 3 n n (3+ ( 3 + ( Chapter revew a Let the vetor e a x + y omponent: x os( a θ 0 os( omponent: y sn( a θ 0 sn(40.9 a Let the vetor e a x + y omponent: x os( a θ 9 os( omponent: y sn( a θ 9 sn( a Let the two vetors e a and, and r a 0os(40 + 0sn( os( sn( r

13 Vetors MC Qld-3 6 r.74 + ( tan(θ.74 θ. The resultant vetor has magntude 3.3 n a dreton of. south of east. 3 a 3+, a ( 3 + ( ( and n + 3 are perpendular ( ( n n ( 3 0 3n n 0 n 0 5 Let the three fores e, and 3 and the resultng fore e R so R os(30 + 5sn( os( sn( os(40 + 0sn( R ( ( R ( (.6 7 N.6 tan(θ (3 rd quadrant or 3.9 south of west The resultng vetor has magntude 7N n a dreton 3.9 south of west. 6 a a sn( θ n ˆ 7 sn(00 nˆ 8.7nˆ 8.7 out of the page. 7 a ( + + ( ( 3 8 ( ( ( ( ( 4 (8 6 + ( A (,, 0, B (3,,, C (4,, AB (3 + ( + ( BC (4 3 + ( + ( 4+ A a AB BC AB BC ( + 4 ( + ( So A ( square unts r + Torque r ( + ( ( + ( (6 0 (6 0 + ( So torque ( Nm 0 60os( sn( OP 0.6os( sn(

14 MC Qld-3 6 Vetors Torque r OP ( ( ( ( ( So torque 3. ( Nm Let ˆn e a unt vetor perpendular to a and If a + and So ˆn a a a ( + ( ( So ˆn ( ( ( 4 + ( 5 ( ( Show that a ( a a where a +, 3 and LHS a ( (6 0 ( ( LHS a ( ( + ( ( + 4 ( (4 6 8 RHS a a ( ( + ( 3 ( ( 3 + ( 0 ( + ( 3 ( ( ( + ( 3 ( 6( 4( 3 + 4( LHS, as requred. 3 a a + +, +, a (8+ 3 (4 + + ( a +,, a (6 + (0 + ( a + +, 3 +, 3 + n+ 4 a n n n 0 (8 0 ( 0 + (3n n 0 6n 8 0 n 8 6 n Let the orgn e at the ottom left-hand orner wth the x-, y- and z-axes defned as usual. V a

15 Vetors MC Qld-3 63 Let a 7 5sn(0 + 5os( os(80 sn( a [ (.6658] 59 u unts 6 The vetors are oplanar. 7 P (, 0,, Q ( 3,.3, R (, 4, 0 PQ ( 3 + ( 0 + (3 5 + QR ( 3 + (4 + ( Let n e a normal to the plane ontanng P, Q and R. n PQ QR (3 5 (5 5 + ( n (x a + n (y + n 3 (z 0 (x 0(y 0 0(z 0 x + 4 0y 0z x 0y 0z x 0y 0z 44 x + 5y + 0z 8 n + +, pont (0, 3, n (x a + n (y + n 3 (z 0 (x 0 + (y 3 + (z 0 x + y 3 + z 0 x + y + z 5 0 x + y + z 5 9 x y + z 6 Let n e a normal to the plane and ˆn a normal unt vetor. n + ˆn n n + + ( + ( ( 6 + Modellng and prolem solvng a The magntudes of the horzontal omponents of the two fores are equal and opposte so T L os(30 T S sn(45 TS os(30 TL sn(45. T S 600N TS T. L T S T L. 600N. 490N W d os( θ where 80N, d 6m and θ 3 So W 80 6 os(3 407N 3 a Let the orgn e at the pvot of the seesaw wth the x- and y-axes taen as usual and the postve z-axs out of the page. OA. S 800 Torque r OA S. ( Nm Steve s weght exerts a torque of 960 Nm Torque 960 Nm A 650 Let OB d where d s Andrea s dstane from the entre (orgn when she provdes a torque equal (and opposte to Steve s. Torque r Torque OB S 960 d ( d 960 ( 650 d d d.48 m Andrea needs to e.48 m from the entre. 4 Let the orgn e at A wth the x-, y- and z-axes taen as usual. AB AC 3 + A a AB AC ( (3

16 MC Qld-3 64 Vetors ( ( (0 + ( ( So A ( 6 + ( m 5 Note the dagram elow s not to sale Let the postve dreton of the x-axs e along vetor OE and the postve dreton of the y-axs e along vetor OA. The postve dreton of the z-axs s perpendular to the x- axs and along the ground as shown n the dagram elow AC AO+ OC OA + OE + EC AB AC Therefore ( ( ( A m 6 a If a p, a p If a 4. a a ( a 4 The magntude remans onstant. A lowse orderng of a, and gves the same answer. An antlowse orderng of a, and gves the negatve of a. 7 m 3 v + r 3 L r mv ( 3 3( + ( 3 ( A a AB AC Pont A s dretly aove pont O, pont B s dretly aove pont D and pont C s dretly aove pont E so OA 3 DB.5 EC And from the aove dagram OE 0 OD 0os(60 + 0sn( AB AO+ OB OA + OD + DB ( Let AB,, AC AD d, et AC + CE AE AC + CB AE 3 ( + + AC CA AB AE 3 ( + AC AB AC AE 3 + ( 3 e e e []

17 Vetors MC Qld-3 65 Let AE dvde BD n a rato m:n m BI BD m+ n m ( d m+ n m m d + m+ n m+ n m ( m+ m+ n d + m+ n m+ n m n d + m+ n m+ n m n m+ n 3 + m+ n But les on the lne AE, therefore we an wrte or e AL AE Susttutng [] and []: m n + m+ n 3 m+ n n Equatng oeffents of results n m+ n 3 3n (m + n [3] Equatng oeffents of results n m 3 m+ n 3 m (m + n [4] [3] + 3 [4] : 3n + 3 m (m + n + 3 (m + n 6m + 6n 3(m + n + 4(m + n 6(m + n 3(m + n + 4(m + n Therefore 6 7 e or We have ust shown that 6 CG C 7 6 AI 7 AE 6 7 e 7 6e CG 6 7 C 7CG 6C 7( g 6( f 7g 7 6 f 6 7g 6f AI AE 7 6 AI AE. In a smlar fashon 7 [] (4+ (7 Ths means that g AG AL But AI AG + GI AG AI + GI So AI GI AI GI AI 6 7 AE 3 7 AE 3 We have ust shown that GI 7 AE. In a smlar fashon 3 GH C 7 The area of ABC The area of GHI GI GH 3 AE 3 C ( 7 7 e f the area of ABC 7