Tiêu đề Vector Varsity Daily-2 (Set-A) Solution.pdf ✅

2 e ̈vL ̈v: Q  †f±iwUi Awfgy‡L P  †f±iwUi AskK = P  . Q  |Q|  q  = 4 + 4 2 2 + 42 Q  |Q|  = 8 4 + 16 2i  + 4k  4 + 16 = 8 20 (2i )  + 4k  = 2 5 (2i )  + 4k  5. hw` A  = 2i  + j  – 3k  I B  = 3i  + 3j  + 3k  nq, Zvn‡j A  I B  Gi ga ̈eZ©x †KvY KZ? [If A  = 2i  + j  – 3k  and B  = 3i  + 3j  + 3k  , what is the angle between A  and B  ?] 90 0 45 60 DËi: 90 e ̈vL ̈v: ABcos = A  .B    = cos–1 6 + 3 – 9 ( 2 ) 2 + 12 + (–3) 2  ( 3 ) 2 + 32 + 32 = 90 Trick : A  .B  = 0 = Ax Bx + Ay By + AzBz n‡j,  = 90 6. (A )   B   C  = 0 n‡j, |C |  . A  = ? [If (A )   B   C  = 0 what is, |C |  . A  = ?] aB  0 – B  C  DËi: 0 e ̈vL ̈v: A   B  = (AB sin)   ; (A )   B   C  = 0 Zvn‡j C  †f±iwU A   B  Gi w`‡Ki mv‡_ GKBw`‡K Ae ̄’vb K‡i| (A )   B  I C  ga ̈eZ©x †KvY 0| A_©vr C  †f±iwU A  †f±iwUi mv‡_ j¤^fv‡e Ae ̄’vb K‡i| ZvB G‡`i WU †cÖvWv‡±i gvb k~b ̈| 7. NÈvq 40 km †e‡M DËi w`‡K Pjgvb GKwU Mvwoi PvjK NÈvq 30 km †e‡M GKwU UavK‡K cwðg w`‡K Pj‡Z †`Lj| UavKwUi cÖK...Z †eM KZ? [A car is moving north at a speed of 40 km/h. The driver sees a truck moving west at a speed of 30 km/h. What is the actual velocity of the truck?] 70 kmh–1 75 kmh–1 50 kmh–1 78 kmh–1 DËi: 50 kmh–1 e ̈vL ̈v: N W E S vT  vc  vT  v  TC VT = V 2 TC + V2 C = 402 + 302 = 50 kmh–1 GLv‡b, Mvwoi cÖK...Z †eM, VC = 40 kmh–1 Mvwoi mv‡c‡ÿ Uav‡Ki †eM, VTC = 30 kmh–1 Uav‡Ki cÖK...Z †eM, VT = ? 8. † ̄avZmn GKwU b`x b~ ̈bZg mg‡q cvi n‡Z PvB‡j, † ̄av‡Zi mv‡_ GKwU †bŠKv‡K KZ †Kv‡Y Pvjv‡Z n‡e? [evqy cÖevn bMY ̈] [A boat needs to cross a river with a minimum time. At what angle should the boat be steered with respect to the current to cross the river in the minimum time? [Neglect wind flow]] 60 120 75 90 DËi: 90 e ̈vL ̈v:     t = d u sin t b~ ̈bZg n‡j, sin †K m‡e©v”P n‡Z n‡e Zvn‡j, sin = 1   = 90 [GLv‡b, t b`x cvi nIqvi mgq, d = `~iZ¡,  = †bŠKv I † ̄av‡Zi ga ̈eZ©x †KvY|] 9. `yBwU mggv‡bi †f±i GKwU we›`y‡Z wμqvkxj| G‡`i jwäi gvb †h‡Kv‡bv GKwU †f±‡ii gv‡bi mgvb n‡j, ga ̈eZ©x †KvY KZ? [Two vectors of equal magnitude act at a point. If the magnitude of their resultant is equal to the magnitude of one of the vectors, what is the angle between them?] 90 120 45 0 DËi: 120 e ̈vL ̈v: P = Q = R n‡j, R = P 2 + Q2 + 2PQ cos  P = P 2 + P2 + 2P2 cos  P 2 = 2P2 + 2P2 cos  –2P2 cos = P2  cos = P 2 – 2P2 = – 1 2 = cos 120   = 120

4 15. wb‡Pi †KvbwU mwjbqWvj Vector Field? [Which of the following is a solenoidal vector field?] DËi: e ̈vL ̈v: mwjbqWv‡ji †ÿ‡Î, †h cwigvY input n‡e †m cwigvY output hv‡e| [Input = Output] 16. P  = 9i  + 27j  + 6k  Q  = 3i  + 9j  + 2k  P  I Q  †f±i mvgvšÍwi‡Ki `yBwU evû n‡j, †ÿÎdj KZ? [If P  = 9i  + 27j  + 6k  Q  = 3i  + 9j  + 2k  are two adjacent sides of a parallelogram, what is the area of the parallelogram?] 0 1 8 12 DËi: 0 e ̈vL ̈v: P   Q  =        i   9 3 j  27 9 k  6 2 = (54 – 54) i  + (18 – 18) j  + (81 – 81)k  = 0 Trick : †h‡nZz Abyiƒc mnM ̧‡jvi AbycvZ mgvb; |P |   Q  = 0 Ges mvgvšÍwi‡Ki †ÿÎdj 0| †Kbbv, 9 3 = 27 9 = 6 2 = 3 Zvn‡j Giv mgvšÍivj, ZvB †ÿÎdj k~b ̈| 17. GKwU e ̄‘‡K 60 N ej Øviv Dˇi Ges 30 N ej Øviv `wÿ‡Y Uvbv n‡”Q| jwä e‡ji gvb KZ? [An object is pulled north with a force of and south with a force of . What is the magnitude of the resultant force?] 40 N 30 N 60 N 90 N DËi: 30 N e ̈vL ̈v: R = P 2 + Q2 + 2PQ cos = (60) 2 + (30) 2 + 2  60  30  cos180 = 30N GLv‡b, P = 60 N Q = 30 N  = 180 R = ? North South 30N 60N 18. GKwU KYvi Dci (i )  – 2j  + mk  | ej cÖ‡qvM Kivq e‡ji w`‡K (2i )  + 3j  – k  m miY nq| e‡ji Øviv m¤úvw`Z Kv‡Ri gvb k~b ̈ n‡j, m Gi gvb KZ? [A particle is acted upon by a force (i )  – 2j  + mk  . The force causes the particle to move in the direction of (2i )  + 3j  – k  m. If the work done by the force is zero, then what is the value of m?] 4 +7 –7 –4 DËi: –4 e ̈vL ̈v: F  .d  = 0  (i )  – 2j  + mk  . (2i )  + 3j  – k  = 0  2 – 6 – m = 0  m = – 4 19. wÎgvwÎK ̄’vbv1⁄4 e ̈e ̄’vq †h‡Kv‡bv †f±‡ii w`K †KvmvBb ̧‡jvi e‡M©i mgwó KZ n‡e? [In a three-dimensional coordinate system, what is the sum of the squares of the direction cosines of any vector?] 0 1 2 3 DËi: 1 e ̈vL ̈v: †h‡Kv‡bv †f±‡ii w`K †KvmvBb ̧‡jvi e‡M©i mgwó me©`v 1 n‡e| A  i  x y j  z   k  