1. 1 Chapter 1: Vectors 2 3

2. unit vector 2 3 3. 3 m , t 4. 4 A v A v A A v 5. 5 - A B= v v A v B v - A B= v v A v B v 6. A v A (Unit vector) A v A v a A v A a A = v A Aa= v a A v 7. 7 x , y z 1 , ,i j k 8. 8 9. 9 BAR += 10. 10 = 275 m 125 m 55.00 228 m 26.70 x ( 1 cm : 10.0 m ) B BAR += R A 11. 11 )( BABA += 12. 13. 13 2 2 x y x x a xa ya a r xa ya x y o , 2 2 x ya a a= + tan y x a a =; a a a iaa xx = jaa yy = jaiaa yx += 14. 14 2 BAC += 15. 15 : 2 145 m 20.0o ( A) 105 m 35.0o ( B). A B C 200 350 y x 16. 16 Vector x component y component A Ax= (145m)sin20.00 = 49.6m Ay= (145m)cos20.00 = 136m B Bx= (105m)cos35.00 = 86.0m By= - (105m)sin35.00 = -60.2m C Cx= Ax + Bx = 135.6 m Cy = Ay + By = 75.8 m mCCC yx 35.155)8.75()6.135( 2222 =+=+= C:C: CC x:x: 011 2.29) 6.135 8.75 (tan)(tan === m m C C x y : 2 C :C : jCiCC yx += 17. 17 3 O P x, y z x, y z 3 r r A * P xA yA zA x z y o x z A y A kAjAiAA zyx ++= xx AA cos= yy AA cos= zz AA cos= 18. cos A Ax x = 222 zAAAA yx ++= r r A * P xA yA zA x z y o x z y 3 kAjAiAA zyx ++= cos A Ay y = A Az z =cos 19. 3 y z x , x, y z 20. Example: A 300 450 3 Ax = 3sin30 cos45 = 3(1/2)(0.707) Ay = 3sin30 sin45 = 3(1/2)(0.707) Az = 3cos30 = 3(0.866) = 1.06 i + 1.06 j + 2.6 k x y z ANS^^ ^ 21. PQ ( ) 2 1r r r = r r r x y Q 2r 1r P r o 2 1x x 2 1y y 1x 2x $ $ $ $ 2 2 2 1 1 1 ( ) ( ) r x i y j z k x i y j z k = + + + + $ $ $ $ 2 1 2 1 2 1( ) ( ) ( )r x x i y y j z z k = + + $ $ $r xi y j zk = + + $ 22. Example: AB, 228 N BA 22 23. 2 2 P Q ( )resultscalarcosPQQP = rr : - , - , PQQP rrrr = ( ) 2121 QPQPQQP rrrrrrr +=+ Example: . 25 x y A B |A| = 50 |B| = 30 A B 24. 000111 ====== ikkjjikkjjii ( ) ( )kQjQiQkPjPiPQP zyxzyx ++++= 2222 PPPPPP QPQPQPQP zyx zzyyxx =++= ++= Example: 2 andkjiAIf 542 += BAdeterminkiB .,63 += 25. 1. 2 : x x y y z z 2 2 2 2 2 2 x y z x y z A B A B A BA B cos( ) A B A A A B B B + + = = + + + + A B A B cos( ) = Example: A B. andkjiAIf 542 += kiB 63 += 26. Example: a. AB AC b. AD AC c. AB AD 27. 2. Projection of a vector : OL OL PP Q QP PQQP OLonPofprojectionPP == = == cos cos cos zzyyxx OL PPP PP coscoscos ++= = For an axis defined by a unit vector: 28. 2 1. 2. 3. V - - ( )BABA = ( ) 2121 BABABBA +=+ sinABBxAC == C C A B 29. 0 0 0 === === === kkikjjki ijkjjkji jikkijii ( ) ( )kBjBiBkAjAiAC zyxzyx ++++= ( ) ( ) ( ) kBABAjBABAiBABA xyyxzxxzyzzy ++= Memory Aid: x y z x y z i j k C A B A A A B B B = = 30. Example: BxAcalculatethen,k5-ji2Bandj4-i3AIf +==