
Position describes a location in space relative to an origin. Coordinates describe a position in three dimensions, using a set of three numbers (such as x, y, and z) and a set of distance units (such as meters). An object’s position is a vector: a quantity with both a value (magnitude) and direction. Displacement is also a vector: It describes a change in position. Vectors can be added graphically (with the “headtotail” rule) or numerically—by separately adding their x, y, and zcomponents.

position, origin, displacement, vector, magnitude, scalar, coordinates

Review problems and questions 


Classify each of the following as a distance or a displacement:
 Jodie sat 20 ft from the teacher’s desk.
 Maurice drove his car a quartermile north of the train station.
 Yasmin hiked 6 km through the San Gabriel Wilderness.
 Ali kicked the ball 40 yards away.

Answer:  Distance: No direction is given.
 Displacement: A direction (“north”) is specified
 Distance: No direction is given.
 Distance: No direction is given.

 In one dimension, positions and displacements can be positive, negative, or zero.
 What is the difference between a negative position and a negative displacement?
 Is it possible to undergo a negative displacement and end up at a positive position?

Answer:  A negative position is any location to the left of the origin. A negative displacement is any change in position that involves moving in the negative direction.
 Yes, it is possible to undergo a negative displacement and end up at a positive location. For example, an object at position +8.0 m can undergo a displacement d = −2.0 m and end up at the position +6.0 m.

 A dog initially located at position (3, 3) m in the x–y plane undergoes the following displacements: d_{1} = (2, 1) m, d_{2} = (−4, −2) m, and d_{3} = (3, −2) m.
 What is the total displacement of the dog?
 What is the final position of the dog with respect to the origin?

Answer:  The total displacement is (+1, −3) m.
 The final position with respect to the origin is (+4, 0) m.

 A plane starts at airport A and undergoes the following displacements before landing at airport B: d_{1} = (+14 km, +32 km) and d_{2} = (+18 km, −11 km). Next, the plane returns directly to airport A.
 What displacement does the plane undergo during the return trip?
 What is the total displacement of the plane during the entire round trip?

Answer:  The displacement from airport B back to A is (−32 km, −21 km).
 The total displacement is zero.

 Step i: A ball rolls from the origin on a onedimensional coordinate axis and stops at (x = 30 m).
Step ii: It next rolls from (x = 30 m) to (x = −30 m).
Step iii: Finally, it rolls from (x = −30 m) to (x = 0 m).
 What is the ball’s displacement during Step i?
 What is the ball’s displacement during Step ii?
 What is the ball’s displacement during Step iii?
 What is the total of the three displacements?
 Generalize about the displacement of any round trip (one that begins and ends at the same position).

Answer:  Displacement = +30 m because the ball moves 30 m in the +xdirection.
 Displacement = −60 m because the ball moves 60 m in the −xdirection.
 Displacement = +30 m because the ball moves 30 m in the +xdirection a second time.
 The displacements add up to zero since (+30 m) + (−60 m) + (+30 m) = 0 m.
 The total displacement is zero for any round trip because your final and original coordinates are identical.

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