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  1. Write each combination of directed line segments as a single directed line segment.
    1. PQ+QR
    2. RP+PS
    3. QSPS
    4. RS+SP+PQ
  2. Given A23 and B21 , find a vector a with representation given by the directed line segment AB . Draw AB and the equivalent representation starting at the origin.
  3. Find the sum of vectors 012 and 003 and illustrate geometrically.
  4. Find a+b , 2a+3b , a , and ab , where a=2i4j+4k and b=2jk .
  5. Find a unit vector with the same direction as 8ij+4k .
  6. The two forces F1 and F2 with magnitudes 10 lb and 12 lb respectively, act on an object at a point P as shown in the figure. Find the resultant force F acting at P as well as its magnitude and its direction. (Indicate the direction by finding the angle θ shown in the figure.)

    three forces

    F=6352i+6+52j ,
    F13.5 lb,
    θ76° .
  7. Velocities have both direction and magnitude and thus are represented by vectors. The magnitude of a velocity is called speed. Suppose that a wind is blowing from the direction N 45° W at a speed of 50 km/h. (This means that the direction from which the wind blows is 45° west of northerly direction.) A pilot is steering a helicopter in the direction N 60° E at an airspeed (speed in still air) of 80 km/h. The true course, or track, of an aircraft is the direction of the resultant of the velocity vectors of the aircraft and the wind. The ground speed of the aircraft is the magnitude of the resultant. Find the true course and the ground speed of the helicopter.
  8. Chains 3 and 5 m in length are fastened to a suspended decoration. The decoration has a mass of 5 kg. The chains, fastened at different heights, make angles of 52° and 40° with the horizontal. Find the tension in each chain and the magnitude of each tension.
  9. The tension at each end of the rope has magnitude 25 N. What is the weight of the rope?

    Let T1 and T2 be the tensions. By symmetry, their horizontal components cancel and their vertical components are equal. The magnitudes of vertical components of T1 and T2 add up to 2T1sin37° . Since the chain is at rest, the force T1+T2 must have the same magnitude as that of the force due to gravity, mg , where g=9.8 . So we can write

    2T1sin37°=9.8m , and hence
    m=50sin37°9.8 ,
    m3.07 [kg].

  10. Use vectors to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.