Chapt_07 - Part II Vectors, Matrices, and Vector Calculus 7...

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Part II Vectors, Matrices, and Vector Calculus 7 7 Vectors EXERCISES 7.1 Vectors in 2-Space 1. (a) 6 i +12 j (b) i +8 j (c) 3 i (d) 65 (e) 3 2. (a) h 3 , 3 i (b) h 3 , 4 i (c) h− 1 , 2 i (d) 5 (e) 5 3. (a) h 12 , 0 i (b) h 4 , 5 i (c) h 4 , 5 i (d) 41 (e) 41 4. (a) 1 2 i 1 2 j (b) 2 3 i + 2 3 j (c) 1 3 i j (d) 2 2 / 3 (e) 10 / 3 5. (a) 9 i +6 j (b) 3 i +9 j (c) 3 i 5 j (d) 3 10 (e) 34 6. (a) h 3 , 9 i (b) 4 , 12 i (c) h 6 , 18 i (d) 4 10 (e) 6 10 7. (a) 6 i +27 j (b) 0 (c) 4 i +18 j (d) 0 (e) 2 85 8. (a) h 21 , 30 i (b) h 8 , 12 i (c) h 6 , 8 i (d) 4 13 (e) 10 9. (a) h 4 , 12 i−h− 2 , 2 i = h 6 , 14 i (b) h− 3 , 9 5 , 5 i = h 2 , 4 i 10. (a) (4 i +4 j ) (6 i 4 j )= 2 i j (b) ( 3 i 3 j ) (15 i 10 j 18 i +7 j 11. (a) (4 i 4 j ) ( 6 i j )=10 i 12 j (b) ( 3 i +3 j ) ( 15 i +20 j )=12 i 17 j 12. (a) h 8 , 0 i−h 0 , 6 i = h 8 , 6 i (b) h− 6 , 0 0 , 15 i = h− 6 , 15 i 13. (a) h 16 , 40 4 , 12 i = h 20 , 52 i (b) h− 12 , 30 10 , 30 i = h− 2 , 0 i 14. (a) h 8 , 12 10 , 6 i = h− 2 , 6 i (b) h− 6 , 9 25 , 15 i = h− 31 , 24 i 339
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7.1 Vectors in 2-Space 15. −−−→ P 1 P 2 = h 2 , 5 i 16. P 1 P 2 = h 6 , 4 i 17. P 1 P 2 = h 2 , 2 i 18. P 1 P 2 = h 2 , 3 i 19. Since P 1 P 2 = −−→ OP 2 1 , 2 = P 1 P 2 + 1 =(4 i +8 j )+( 3 i +10 j )= i +18 j , and the terminal point is (1 , 18). 20. Since P 1 P 2 = 2 1 , 1 = 2 P 1 P 2 = h 4 , 7 i−h− 5 , 1 i = h 9 , 8 i , and the initial point is (9 , 8). 21. a (= a ), b (= 1 4 a ), c (= 5 2 a ), e (= 2 a ), and f (= 1 2 a ) are parallel to a . 22. We want 3 b = a ,so c = 3(9) = 27. 23. h 6 , 15 i 24. h 5 , 2 i 25. k a k = 4+4=2 2; (a) u = 1 2 2 h 2 , 2 i = h 1 2 , 1 2 i ; (b) u = h− 1 2 , 1 2 i 26. k a k = 9+16=5; (a) u = 1 5 h− 3 , 4 i = h− 3 5 , 4 5 i ; (b) u = h 3 5 , 4 5 i 27. k a k =5; (a) u = 1 5 h 0 , 5 i = h 0 , 1 i ; (b) u = h 0 , 1 i 28. k a k = 1+3=2; (a) u = 1 2 h 1 , 3 i = h 1 2 , 3 2 i ; (b) u = h− 1 2 , 3 2 i 29. k a + b k = kh 5 , 12 ik = 25 + 144 = 13; u = 1 13 h 5 , 12 i = h 5 13 , 12 13 i 30. k 2 a 3 b k = kh− 5 , 4 ik = 25+16= 41; u = 1 41 h− 5 , 4 i = h− 5 41 , 4 41 i 31. k a k = 9+49= 58; b =2( 1 58 )(3 i +7 j 6 58 i + 14 58 j 32. k a k = q 1 4 + 1 4 = 1 2 ; b =3( 1 1 / 2 )( 1 2 i 1 2 j 3 2 2 i 3 2 2 j 33. 3 4 a = h− 3 , 15 / 2 i 34. 5( a + b )=5 h 0 , 1 i = h 0 , 5 i 35. 36. 37. x = ( a + b a b 38. x a b )=2 a 2 b 340
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7.1 Vectors in 2-Space 39. b =( c ) a ;( b + c )+ a = 0 ; a + b + c = 0 40. From Problem 39, e + c + d = 0 . But b = e a and e = a + b ,so( a + b c + d = 0 . 41. From 2 i +3 j = k 1 b + k 2 c = k 1 ( i + j k 2 ( i j )=( k 1 + k 2 ) i +( k 1 k 2 ) j we obtain the system of equations k 1 + k 2 =2, k 1 k 2 = 3. Solving, we ±nd k 1 = 5 2 and k 2 = 1 2 . Then a = 5 2 b 1 2 c . 42. From 2 i j = k 1 b + k 2 c = k 1 ( 2 i +4 j k 2 (5 i +7 j 2 k 1 +5 k 2 ) i +(4 k 1 k 2 ) j we obtain the system of equations 2 k 1 k 2 =2,4 k 1 k 2 = 3. Solving, we ±nd k 1 = 1 34 and k 2 = 7 17 . 43. From y 0 = 1 2 x we see that the slope of the tangent line at (2 , 2) is 1. A vector with slope 1 is i + j . A unit vector is ( i + j ) / k i + j k i + j ) / 2= 1 2 i + 1 2 j . Another unit vector tangent to the curve is 1 2 i 1 2 j .
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This note was uploaded on 10/24/2011 for the course ACAD 111 taught by Professor 123a during the Spring '11 term at Aachen University of Applied Sciences.

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Chapt_07 - Part II Vectors, Matrices, and Vector Calculus 7...

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