If you take two steps of different sizes, can you end up at your starting point? More generally, can two vectors with different magnitudes ever add to zero? Can three or more?
Use graphical methods to solve these problems. You may assume data taken from graphs is accurate to three digits.
Find the following for path A in
[link] : (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.
Find the following for path B in
[link] : (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.
Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements
and
, as in
[link] , then this problem asks you to find their sum
.)
Suppose you first walk 12.0 m in a direction
west of north and then 20.0 m in a direction
south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements
and
, as in
[link] , then this problem finds their sum
.)
Repeat the problem above, but reverse the order of the two legs of the walk; show that you get the same final result. That is, you first walk leg
, which is 20.0 m in a direction exactly
south of west, and then leg
, which is 12.0 m in a direction exactly
west of north. (This problem shows that
.)
(a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction
north of east (which is equivalent to subtracting
from
—that is, to finding
). (b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction
south of west and then 12.0 m in a direction
east of south (which is equivalent to subtracting
from
—that is, to finding
). Show that this is the case.
Show that the
order of addition of three vectors does not affect their sum. Show this property by choosing any three vectors
,
, and
, all having different lengths and directions. Find the sum
then find their sum when added in a different order and show the result is the same. (There are five other orders in which
,
, and
can be added; choose only one.)
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