Supplementary Problems

Vector quantities can be added according to the following rule: Each vector can be represented by an arrow and is properly shown when the length of the arrow shows the size of the vector while the direction of the arrow shows the direction of the vector as measured against some reference direction. The vectors can be added by linking them head to tail, taking care to ensure that their relative size and direction remain the same during thee linking The vector sum of the vector is the length of the segment drawn from the tail of the first vector to the head of the last. This vector sum is called a resultant vector. Before the sum of the vectors can be computed, each vector has to be separated into x- and y-components.This is called resolving a vector. For each vector create a right trangle where the vector is the hypotenuse and the legs are parallel to an appropriate pair of x-y axes. The vector components will be the product of the magnitude of the vector times the sine or cosineof the angle. Do this for all of the vectors to be added. Add all the z--componnts; add all the y-components. Finally, combine these two sums using the Pythagorean theorem.

Note that each of these problems can be solved by vector addition as defined above by following this protocol

a) Draw a diagram scaled as best you can.
b) Install a useful coordinate system.
c) Resolve the vector in the x and y directions
e) Use the Pythagorean formula to calculate the length of the hypotenuse of this triangle. The length of this segment is the sum of the vectors
f) Use some trig function to determine the direction of the vector sum. This direction may be some number of degrees, from north, from the edge of a river bank, from horizontal or vertical, or as measured from some other reference point.

1. Jill walks 200 m @ 37o. She then turns and walks 300 m @ 112o. Jack, that rascal, wants to join Jill by the shortest possible route. a) How far should he plan to walk? b) In what direction should he walk? c) In what direction should they travel to return home by the shortest possible route?

2. In problem #1, what distance and direction will Jack walk if Jill walks these two displacements in a different order. Draw the vector diagram and set up the math. What do you notice?

3. An airplane has an air speed (that is, speed of plane with respect to air) of 400 mi/hr @ 200o. It encounters a prevailing wind of 100 mi/hr @ 250o (speed of air with respect to ground). What is the speed of the plane with respect to the ground?

4. Stan and Ollie have attached ropes to a box resting on the floor. Stan can muster 500 N @ 115o. Ollie can pull 400 N @ 180o. What is the vector sum of these forces? A student wishes to apply a third force that will cancel out the other two forces (this is called an equilibrant force). How big should this force be? In what direction should it point?

5. Two forest fire watch towers are 10 miles apart and lie on an east-west parallel.  An observer on the western tower observes a puff of smoke at a bearing of 15o.  The eastern tower reports  a fire at a bearing of 330o How far is the fire from each tower?

6. Jack and Jill are walking together this time. They have an appointment at 30 units of distance and at a bearing 47o. Being the man that he is, Jack takes control of the situation. At his insistence, they walk 10 units at 170o, then turn and walk20 units at . at the end of this trek, jack does an unnatural act, he asks directions from you.What directions do you give him to get from where he is now to where he wants to be (assuming that he can and will follow a straight line path to get there