Content text Vector Engineering Practice Sheet Solution (HSC 26).pdf
4 Higher Math 1st Paper Chapter-2 16| AB = 3i ^ + 2j ^ – k ^ Ges AC = 5i ^ – j ^ + 2k ^ n‡j, AB I AC †K mwbœwnZ evû a‡i AswKZ mvgvšÍwi‡Ki †ÿÎdj wbY©q Ki| [RUET 13-14; BUET 09-10, 04-05] mgvavb: AB AC = i ^ 3 5 j ^ 2 – 1 k ^ – 1 2 = 3i ^ – 11j ^ – 13k ^ AB I AC †K mwbœwnZ evû a‡i AswKZ mvgvšÍwi‡Ki †ÿÎdj, |AB AC | = 3 2 + (11) 2 + (– 13) 2 = 299 (Ans.) weMZ mv‡j CUET-G Avmv cÖkœvejx 17| aaæeK a Gi gvb wbY©q Ki †hb 2i ^ + j ^ – k ^ , 3i ^ – 2j ^ + 4k ^ , i ^ – 3j ^ + ak ^ †f±i wZbwU GKB mgZ‡j _v‡K| [CUET 07-08] mgvavb: 2 3 1 1 –2 –3 –1 4 a = 0 – 4a + 24 – 3a + 4 + 7 = 0 a = 5 (Ans.) 18| †f±i c×wZ‡Z GKwU wÎfz‡Ri †ÿÎdj wbY©q Ki, hvi kxl©we›`yÎq h_vμ‡g A(1, 3, 2), B(2, – 1, 1) Ges C(– 1, 2, 3). [CUET 04-05] mgvavb: OA = i ^ + 3j ^ + 2k ^ , OB = 2i ^ – j ^ + k ^ , OC = – i ^ + 2j ^ + 3k ^ GLb, AB = OB – OA = i ^ – 4j ^ – k ^ , BC = OC – OB = – 3i ^ + 3j ^ + 2k ^ ABC = 1 2 |AB BC | AB BC = i ^ 1 –3 j ^ –4 3 k ^ –1 2 = – 5i ^ + j ^ – 9k ^ ABC = 1 2 25 + 1 + 81 = 1 2 107 (Ans.) weMZ mv‡j BUTex-G Avmv cÖkœvejx 19| †`LvI †h, A = 8i ^ + j ^ – 6k ^ Ges B = 4i ^ – 2j ^ + 5k ^ †f±i `yBwU ci ̄úi j¤^| [BUTex 07-08] mgvavb: A . B = (8i ^ + j ^ – 6k ^ ).(4i ^ – 2j ^ + 5k ^ ) = 8 4 + 1 (– 2) + (– 6) 5 = 0 A Ges B †f±iØq ci ̄úi j¤^| (Showed) 20| A I B we›`yi Ae ̄’vb †f±i h_vμ‡g, 2i ^ + 3j ^ – 4k ^ I 4i ^ – 3j ^ + 2k ^ | AB Gi gvb Ges AB eivei GKK †f±i wbY©q Ki| [BUTex 06-07] mgvavb: AB = OB – OA = (4i ^ – 3j ^ + 2k ^ ) – (2i ^ + 3j ^ – 4k ^ ) = 2i ^ – 6j ^ + 6k ^ |AB | = 4 + 36 + 36 = 76 (Ans.) awi, AB eivei GKK †f±i = a ^ a ^ = AB |AB | = 2i ^ – 6j ^ + 6k ^ 76 (Ans.) 21| †`LvI †h, r = i ^ + j ^ + k ^ †f±iwU Aÿ·qi mv‡_ mgvb †Kv‡Y AvbZ| [BUTex 05-06] mgvavb: awi, x Aÿ eivei GKK †f±i, a = i ^ GLb, | a| = 1 r = i ^ + j ^ + k ^ Ges |r | = 1 2 + 12 + 12 = 3 Avevi, cos = r .a |r | |a | cos = 1 1 3 1 = 1 3 = cos–1 1 3 Abyiƒcfv‡e, y I z A‡ÿi †ÿ‡Î cÖgvY Kiv hvq, = cos–1 1 3 (Showed) 22| A = 3i ^ – 2j ^ + k ^ , B = i ^ – 3j ^ + 5k ^ I C = 2i ^ + j ^ – 4k ^ †f±img~n Øviv GKwU mg‡KvYx wÎfzR ˆZwi Kiv wK m¤¢e? [BUTex 03-04] mgvavb: GLv‡b, B + C = i ^ – 3j ^ + 5k ^ + 2i ^ + j ^ – 4k ^ = 3i ^ – 2j ^ + k ^ = A A , B Ges C wÎfzR MVb K‡i| GLb, |A | 2 = 9 + 4 + 1 = 14 Avevi, |B | 2 = 1 + 9 + 25 = 35 Avevi, |C | 2 = 4 + 1 + 16 = 21 |B | 2 = |A | 2 + |C | 2 = 35 Avevi, A . C = 6 – 2 – 4 = 0 †h‡nZz †f±i ؇qi DU ̧Ydj k~b ̈ Ges B 2 = A2 + C2 ; mg‡KvYx wÎfzR MVb Kiv m¤¢e| (Ans.)
Tiếng Việt