B Apply

Let u = [ − 5 3 ] , v = [ 4 − 3 ] , u equals . matrix with 2 rows and 1 column , row1 column 1 , negative 5 , row2 column 1 , 3 , end matrix . comma . v equals . matrix with 2 rows and 1 column , row1 column 1 , 4 , row2 column 1 , negative 3 , end matrix . comma  And w = [ 2 2 ] . w equals , matrix with 2 rows and 1 column , row1 column 1 , 2 , row2 column 1 , 2 , end matrix . .  Find the following vectors.

32. 2u + 3v
33. 2 v − 4 w 2 v minus 4 w
34. − u − w negative u minus w
35. − 3 u + v − 1 2 w negative 3 u plus v minus , 1 half , w
36. Think About a Plan A ferry shuttles people from one side of a river to the other. The speed of the ferry in still water is 25 mi/h. The river flows directly south at 7 mi/h. If the ferry heads directly west, what is the ferry's resulting speed?
    • How can a sketch help you solve this problem?
    • What formula can you use to find the speed?
37. Aviation A twin-engine airplane has a speed of 300 mi/h in still air. Suppose the airplane heads south and encounters a wind blowing 50 mi/h due east. What is the resultant speed of the airplane?
38. Aviation A small airplane lands at a point 216 mi east and 76 mi north of the point from which it took off. How far did the airplane fly?
39. Consider the triangle with vertices at A(2, 2), B(5, 3), and C(3, 6). Express the sides of the triangle as vectors A B ⇀ , B C ⇀ , modified eh b with right harpoon up above , comma , modified b cap c with right harpoon up above , comma  and C A ⇀ . modified cap c eh with right harpoon up above , .

Let a = 〈6, − 1 negative 1 〉, b = 〈 − 4 , negative 4 comma  3〉, and c = 〈2, 0〉. Solve each of the following for the unknown vector v.

40. a + v = b
41. c − v = b c minus v equals b
42. v − b = a + c v minus b equals eh plus c
43. a + b + c + v = (0, 0)
44. Navigation A fishing boat leaves its home port and travels 150 mi directly east. It then changes course and travels 40 mi due north. How long will the direct return trip take if the boat averages 23 mi/h?
45. Writing Subtract any vector from itself. The result is still a vector, but a unique one. Explain what this vector is, and what it means for vector addition.

Reasoning Do the following properties hold for vectors and scalars? Identify each property and make a diagram to support your answers.

46. u + v u plus v = v + u
47. k( u + v u plus v ) = k u + k v
48. u − v = v − u u minus v equals v minus u
49. ( u + v u plus v ) + w = u + (v + w)

C Challenge

50. Aviation A helicopter starts at (0, 0) and makes three legs of a flight represented by the vectors 〈10, 10〉, 〈5, − 4 negative 4 〉, and 〈 − 3 , 5 〉 , negative 3 comma 5 right pointing angle bracket comma in that order. If another helicopter starts at (0, 0) and flies the same three legs in a different order, would it end in the same place? Justify your answer.
51. Two vectors are parallel if the absolute value of their dot product is equal to the product of their magnitudes. Which of the following vectors are parallel? Which are perpendicular?

    A = [ 0.9 1.2 ] eh equals . matrix with 2 rows and 1 column , row1 column 1 , 0.9 , row2 column 1 , 1.2 , end matrix

    b = [ − 2 1.5 ] b equals . matrix with 2 rows and 1 column , row1 column 1 , negative 2 , row2 column 1 , 1.5 , end matrix

    c = [ 6 − 8 ] c equals . matrix with 2 rows and 1 column , row1 column 1 , 6 , row2 column 1 , negative 8 , end matrix

    d = [ − 4.5 − 6 ] d equals . matrix with 2 rows and 1 column , row1 column 1 , negative 4.5 , row2 column 1 , negative 6 , end matrix

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