40. Think About a Plan A Red Cross helicopter takes off and flies to deliver some relief supplies. Then the helicopter flies to another location to pick up three nurses. The helicopter's flight path is shown. What is the helicopter's distance from its point of origin? Round to the nearest kilometer.

    

    • What information does the diagram give you?
    • What do you know about the angle formed by the two parts of the trip?

Give the sum of a ⇀ modified a with right harpoon up above and b ⇀ . modified b with right harpoon up above , . Draw a ⇀ modified a with right harpoon up above and b ⇀ modified b with right harpoon up above and their sum in the coordinate plane.

41. a ⇀ = 〈 − 5 , − 2 〉 , b ⇀ = 〈 2 , − 5 〉 modified a with right harpoon up above , equals left pointing angle bracket negative 5 comma negative 2 right pointing angle bracket comma , modified b with right harpoon up above , equals , left pointing angle bracket 2 comma , minus 5 right pointing angle bracket
42. a ⇀ = 〈 5 , − 2 〉 , b ⇀ = 〈 − 5 , − 2 〉 modified a with right harpoon up above , equals , left pointing angle bracket 5 comma , minus 2 right pointing angle bracket comma , modified b with right harpoon up above , equals left pointing angle bracket negative 5 comma negative 2 right pointing angle bracket
43. a ⇀ = 〈 5 , − 5 〉 , b ⇀ = 〈 − 2 , 2 〉 modified a with right harpoon up above , equals , left pointing angle bracket 5 comma . negative 5 right pointing angle bracket comma . modified b with right harpoon up above , equals . left pointing angle bracket negative 2 comma 2 right pointing angle bracket
44. Open-Ended Name four other vectors with the same magnitude as 〈 − 7 , − 24 〉 . left pointing angle bracket negative 7 comma . negative 24 right pointing angle bracket .

45. Use u ⇀ = 〈 2 , 4 〉 , v ⇀ = 〈 3 , 1 〉 , modified u with right harpoon up above , equals . left pointing angle bracket 2 comma 4 right pointing angle bracket comma . modified v with right harpoon up above , equals . left pointing angle bracket 3 comma 1 right pointing angle bracket comma and w ⇀ = 〈 5 , − 1 〉 . modified w with right harpoon up above , equals , left pointing angle bracket 5 comma . negative 1 right pointing angle bracket .

    1. Find u ⇀ + v ⇀ modified u with right harpoon up above , plus , modified v with right harpoon up above and v ⇀ + u ⇀ . modified v with right harpoon up above , plus , modified u with right harpoon up above , .
    2. Find u ⇀ + ( v ⇀ + w ⇀ ) modified u with right harpoon up above , plus . open . modified v with right harpoon up above , plus , modified w with right harpoon up above . close and ( u ⇀ + v ⇀ ) + w ⇀ . open . modified u with right harpoon up above , plus , modified v with right harpoon up above . close . plus , modified w with right harpoon up above , .
    3. Make a Conjecture Which properties of real number addition seem to hold true for vector addition?
    4. Prove your conjecture algebraically and geometrically.

46. Aviation An airplane takes off from a runway in the direction 10 ° 10 degrees east of south. When it reaches 5000 ft, it turns right 45 ° . 45 degrees . It cruises at this altitude for 60 mi. Then it turns left 160 ° , 160 degrees comma descends, and lands. Match each vector with the appropriate portion of the flight.

    The plane takes off. The plane cruises. The plane lands.

47. Aviation The cruising speed of one model of commercial jet in still air is 530 mi/h. Suppose that this jet is cruising directly east when it encounters an 80 mi/h wind blowing 40 ° 40 degrees south of west.

    1. Sketch the vectors for the velocities of the airplane and the wind.
    2. Express both vectors from part (a) in ordered pair notation.
    3. Find the sum of the vectors from part (b).
    4. Find the magnitude and direction of the vector from part (c).

48. Navigation A fishing boat leaves its home port and travels 150 mi directly east. It then changes course and travels 40 mi due north.

    1. In what direction should the boat head to return to its home port?
    2. If the boat averages 23 mi/h, how long will the return trip take?

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