u v u v v 2 v u 5, 12, v 3, 2 3. u v u 3i 4j, v 7i 2j u v u 4i 2j, v i j 6. u v u v u i 2j, v 2i j 9.

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1 Section. Vectors and Dot Prodcts 53 Vocablary Check 1. dot prodct. 3. orthogonal. \ 5. proj PQ F PQ \ ; F PQ \ 1., 1,, 3. 5, 1, 3, 3., 1,, , 5, 1, 5. i j, i j. 3i j, 7i j i j, i 3j 8. i j, i j 9., The reslt is a scalar. 10.,, 3, The reslt is a scalar.,, 3, 3 3, 3,, 8 The reslt is a ector. 1.,, 3,, w 1, 13. w 3 1, 1,, ector,, 3,, w 1, 3w 313 3, 33,, The reslt is a ector. 1.,, 3,, w 1, 15. w 1,, 8 w w 81, The reslt is a scalar. 1,, 8 ector 1., 8 17.,, 3,, w 1, w 3 1 scalar The reslt is a scalar.

2 5 Chapter Additional Topics in Trigonometry 18.,, 3,, w 1, w scalar 19. 5, , i 5j i 1j j i , 0, 0,. 3,,, i j, j cos cos cos arccos i 3j, i j 9. cos i j, i j cos i 3j, 8i j 31. cos i 5j, i j cos i 3j, i 3j cos

3 Section. Vectors and Dot Prodcts cos cos 3 i 3 sin 1 3 i sin cos 1 3 arccos 3 j 1 3 i j 3. j i j cos cos cos i sin i sin j j j i j i j y 3. 7i 5j cos x i 3j, i j cos 3 3 cos 1 10 cos cos 1 10 y cos cos x 37. 5i 5j y 38. 8i 8j cos x i 3j, 8i 3j cos cos cos 7 cos cos 7.9 y 8 10 x

4 5 Chapter Additional Topics in Trigonometry 39. P 1,, Q 3,, R, 5 PQ \,, PR \ 1, 3, QR \ 1, 1 PR \ cos PQ\ PQ \ PR \ 8 10 arccos.57 5 QR \ cos PQ\ Ths, PQ \ QR \ P 3,, Q 1, 7, R 8, 1. PQ \, 11, QR \ 7, 5, PR \ 11,, QP \, 11 PR \ cos PQ\ PQ \ PR \ QP \ cos QR\ QR \ QP \ P 3, 0, Q,, R 0, QP \ 5,, PR \ 3,, QR \,, PQ \ 5, PR \ cos PQ\ PQ \ PR \ QR \ cos QP\ QP \ PR \ P 3, 5, Q 1, 9, R 7, 9 3. PQ \,, QR \ 8, 0, PR \ 10,, QP \, PR \ cos PQ\ PQ \ PR \ QP \ cos QR\ QR \ QP \ cos 10 cos , 50, cos cos 5, , cos 93 cos cos 1 cos , 30, 1, 5 8. and are parallel. 3, 15, 1, Not orthogonal Neither k Not parallel 1 3i j, 5i j k Not parallel 0 Not orthogonal Neither

5 Section. Vectors and Dot Prodcts , i j 51. k Not parallel 0 Not orthogonal Neither i j, i j 5. cos, sin 0 and are orthogonal. sin, cos 0 and orthogonal. are 53.,,, 1 w 1 proj 1 1, 1 8, w w 1, 1 37, , , 1 10, 0, , 10, ,, 1, 55. w 1 proj 01, 0, 0 w w 1, 0, 0,, 0, 0, 0, 3,, 15 w 1 proj 5 9 w w 1 0, 3 5 9, , , 9, 15 5, 15 15, 0, ,,, proj 0 since they are perpendiclar. w 1 proj 1, 1 17 w w 1 3, 1 5, 1 1, , 1 1, 3, Since and are orthogonal, 0 and proj 0. proj 0, since Becase and are orthogonal, the projection of onto is 0. proj w 0 since , , 3 For to be orthogonal to, mst eqal 0. For to be orthogonal to, mst be eqal to 0. Two possibilities: 5, 3 and 5, 3 Two possibilities: 3, 8, 3, i 3 j For and to be orthogonal, mst eqal 0. Two possibilities: 3 i 1 j and 3 i 1 j. 5 i 3j For to be orthogonal to, mst be eqal to 0. Two possibilities: 3i 5 and 3i 5 j j

6 58 Chapter Additional Topics in Trigonometry 3. \ w proj PQ PQ\ where PQ \, 7 and 1,.. P 1, 3, Q 3, 5, i 3j proj PQ \ PQ \ PQ \ PQ\ 3, 7 5 work PQ \ i 3j i j \ w proj PQ 35 PQ\ (a) 150, 300, 15.5, $58,7.50 This gies the total reene that can be earned by selling all of the pans. (b) Increase prices by 5%: 1.05 The operation is scalar mltiplication ,7.50 1, (a) 30, 50, 1.75, The fast food stand sold $ of hambrgers and hot dogs in one month. (b) Increase prices by.5%: 1.05 scalar mltiplication 7. (a) Force de to graity: F 30,000j Unit ector along hill: cos di sin dj Projection of F onto : w 1 proj F F F 30,000 sin d (b) The magnitde of the force is 30,000 sin d. d Force (c) Force perpendiclar to the hill when d 5: Force 30, ,885.8 ponds 8. Force de to graity: Unit ector along hill: Projection of F onto : F 500j cos 10i sin 10j w 1 proj F F F becase is a nit ector, 1 0cos sin sin The magnitde of the force is 937.7, so a force of ponds is reqired to keep the ehicle from rolling down the hill. Force perpendiclar to the hill: Force ponds

7 Section. Vectors and Dot Prodcts w newton-meters 70. work 005 1,000 foot-ponds 71. w cos foot-ponds 7. work cos 3515, ,8,5 newton-meters 73. w cos ,50. foot-ponds 7. work cos F PQ \ cos 05 ponds50 feet 117. foot-ponds 75. False. Work is represented by a scalar. 7. Tre. W F PQ \ 0 if F and PQ \ are orthogonal. 77. (a) (b) (c) 0 and are orthogonal and. 78. (a) proj and are parallel. > 0 cos > 0 0 < 0 cos < 0 < < (b) proj 0 and are orthogonal. 79. In a rhombs,. The diagonals are 80. Let 1, and 1,. and. 1 1, 0 Therefore, the diagonals are orthogonal ii i i1 i9 85. i sin x 3 sin x 0 sin x cos x 3 sin x 0 sin x cos x 3 0 sin x 0 or cos x 3 0 x 0, cos x 3 x, 11

8 570 Chapter Additional Topics in Trigonometry 8. sin x cos x tan x tan x sin x cos x cos x 0 cos x sin x 0 cos x 0 x, 3 sin x 0 sin x tan x tan x 1 tan x tan x1 tan x tan x tan x1 tan x tan x 0 tan x1 tan x 1 0 x, 5, 3, 7 x 5, 7 tan xtan x 0 tan 3 x 0 tan x 0 x 0, 88. cos x 3 sin x 1 sin x 3 sin x 0 sin x 3 sin x 1 0 sin x 1sin x 1 0 sin x 1 0 sin x 1 0 sin x 1 x 7, 11 x 7, 3, 11 sin x 1 x 3 For Exercises 89 9: sin 1 13, in Qadrant IV cos 5 13 cos 5, in Qadrant IV sin sin sin cos cos sin sin 1 13, cos cos 5, sin sin sin cos cos sin cos cos cos sin sin sin 1 5, cos, tan cos 5, sin 7, tan 7 5 tan tan tan 1 tan tan

6.4 VECTORS AND DOT PRODUCTS

458 Chapter 6 Additional Topics in Trigonometry 6.4 VECTORS AND DOT PRODUCTS What yo shold learn ind the dot prodct of two ectors and se the properties of the dot prodct. ind the angle between two ectors