Dealing with vectors, Pirate-style

Once upon a time, on a deserted plot of ground, a crew of pirates found 3 colored dots. "This must be the map to buried treasure!" they cried, and carefully measured the positions of each dot.

                X (cm)               Y (cm)
----------------------------------------------------

 1          -11.8 +/- 0.3         9.7 +/- 0.2

 2          -11.4 +/- 0.2        -8.6 +/- 0.4

 3            7.1 +/- 0.3         2.8 +/- 0.3
----------------------------------------------------

The pirates erased all their coordinate markings and left.

Several days later, a clan of ninjas stumbled across the same three colored dots. "Waah! Dots! Let us measure them!" The ninjas created their own coordinate system and made a careful set of measurements.

                X (cm)               Y (cm)
----------------------------------------------------

 1            6.0 +/- 0.2        12.0 +/- 0.3

 2          -10.1 +/- 0.4         2.9 +/- 0.2

 3            8.9 +/- 0.3        -8.1 +/- 0.3
----------------------------------------------------

Now, choose one team: Ninja or Pirate.

Use your team's measurements to compute the components of the vectors which run

• r₁₂ which runs from dot 1 to dot 2
• r₂₃ which runs from dot 2 to dot 3
• r₃₁ which runs from dot 3 to dot 1

For each component of each displacement, calculate the measurement uncertainty, using the error propagation rules. Fill in the table below for your team.

  Table 1:  Dot-to-dot displacement vectors 

                       Pirate                          Ninja

                X (cm)       Y (cm)              X (cm)        Y (cm)
--------------------------------------------------------------------------

 r12           +/-              +/-                +/-            +/- 

 r23           +/-              +/-                +/-            +/- 

 r31           +/-              +/-                +/-            +/- 
--------------------------------------------------------------------------

Now calculate the magnitude of the displacement vector between each pair of points and its uncertainty, using the information recorded above, and enter into a table. Compute the uncertainty in each magnitude as well.

How to calculate the uncertainty in magnitude? Well, consider a right triangle with sides A and B, and hypotenuse C. You can read a detailed description of the uncertainty calculations if you wish; here's a short version.

Pythagoras says that

              C2  =  [ A2  +  B2 ]

The uncertainties in A² and B² are

 
  (      ΔA     )            (      ΔB     )
  ( 2 * ------  ) * A2       ( 2 * ------  ) * B2
  (      A      )            (      B      )

which means that the uncertainty in C² must be the sum:

 
             (      ΔA     )            (      ΔB     )
 Δ(C2) =     ( 2 * ------  ) * A2   +   ( 2 * ------  ) * B2
             (      A      )            (      B      )

• Simplify the above equation for uncertainty in C².
• Use the standard propagation of uncertainty rules to write an equation for the uncertainty in C itself, based on the value and uncertainty in C².
• Now you can fill in the table below with the magnitude and uncertainty of each difference vector.

   Table 2:       Magnitude (cm) of the difference vectors

                        Pirates                     Ninjas 
-----------------------------------------------------------------

 r12                    +/-                          +/-

 r23                    +/-                          +/-

 r31                    +/-                          +/-
-----------------------------------------------------------------

Questions:

1. Compare the vector components (Table 1) you measured, against those of the other team. Are they the same or different, within uncertainty? Explain the result.
2. Compare the vector magnitudes (Table 2) you measured, against those of the other team. Are they the same or different within uncertainty? Explain the result.
3. Suppose we now add your three displacement vectors.
   • What should be the result of this vector sum?
   • Do your results agree with your expectation, within uncertainty?