Elevator down |

**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**

**d) Add the x-components together; and then add the y-components
together**

**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 @ 37 ^{o.} She then turns and
walks 300 m @ 112^{o}. 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 @ 200 ^{o}. It encounters
a prevailing wind of 100 mi/hr @ 250^{o} (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 @ 115 ^{o}. Ollie can
pull 400 N @ 180^{o}. 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 15 ^{o. }The
eastern tower reports a fire at a bearing of 330^{o}
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**