Treasure Map

Long John Silver, a pirate, has buried his treasure on an island with five trees located at the following points: A (30.0 m, –20.0 m), B (60.0 m, 80.0 m), C (–10.0 m, –10.0 m), D (40.0 m, –30.0 m), and E (–70.0 m, 60.0 m). All of the points are measured relative to some origin, as in Figure. Long John’s map instructs you to start at A and move toward B, but cover only 1/2 the distance between A and B. Then move toward C, covering 1/3 the distance between your current location and C. Then move toward D, covering 1/4 the distance between where you are and D. Finally, move toward E, covering 1/5 the distance between you and E, stop, and dig.

A (30.0 m, –20.0 m), B (60.0 m, 80.0 m), C (–10.0 m, –10.0 m)

D (40.0 m, –30.0 m), and E (–70.0 m, 60.0 m).

The intermediate points are F,G,H and I. For each of coordinates x and y:
 F = A + 1/2 (B - A)
 G = F + 1/3 (C - F)
 H= G + 1/4 (D - G)
 I = H + 1/5 (E - H)

Vectors:
F = (30,-20) + 1/2 ((60,80) - (30,-20)) = (30,-20) + 1/2 (30,100) =  (45, 30)

 G = (45,30) + 1/3 ((-10,-10) - (45, 30)) = (45, 30) + 1/3 (-55, -40) = = (45-55/3,30-40/3)
 

H= ... 

or you can use a program

XA=30.0

YA=-20.0

XB=60.0

YB=80.0

XC=-10.0

YC=-10.0

XD=40.0

YD=-30.0

XE=-70.0

YE=60.0

XF=XA+(XB-XA)/2.

YF=YA+(YB-YA)/2.

XG=XF+(XC-XF)/3.

YG=YF+(YC-YF)/3.

XH=XG+(XD-XG)/4.

YH=YG+(YD-YG)/4.

XI=XH+(XE-XH)/5.

YI=YH+(YE-YH)/5.

print *,XI,YI

output: 10. 16.

Treasure Island is an adventure novel by author Robert Louis Stevenson, narrating a tale of "buccaneers and buried gold". Traditionally considered a coming of age story, it is an adventure tale known for its superb atmosphere, character and action, and also a wry commentary on the ambiguity of morality—as seen in Long John Silver—unusual for children's literature then and now. The influence of Treasure Island on popular perception of pirates is vast, including treasure maps with an 'X', schooners, the Black Spot, tropical islands, and one-legged seamen with parrots on their shoulders.

Prodotto esterno di r e v

Domanda interessantissima di uno studente:
“Abbiamo visto studiando il moto circolare che wxr=v (per chi ha Firefox: la omega risulta visualizzata come  w). Questi tre vettori formano una terna di vettori ortogonali. Se guardo la figura, vedo il prodotto rxv parallelo ad w. Le dimensioni di rxv sono [L²/t] e quelle di w sono [1/t]. 

C’è una grandezza relativa al prodotto  rxv in qualche modo legata a w, di modo che rxv=cw  ? Che cosa è c e che dimensioni ha?"

La grandezza esiste e la vediamo più avanti. Ma proviamo a calcolare nel seguente modo. Pensiamo ad una particella di massa m che si muove su una circonferenza con velocita’ angolare w . Calcoliamo il prodotto, ricordando che wxr = v:

Per il moto circolare, c è r² che ha dimensioni [L²]. Più avanti studiamo L vettore che rappresenta il “momento angolare“ ed I il “momento d’inerzia” mr².

Prodotto esterno di vettori

The Cross Product (prodotto esterno di vettori)

The product of two vectors in the space according to the cross or vector product  is a another vector. This new vector is perpendicular to the plane in which the first two vectors lie. Note that there are two possible directions in which the cross product of two vectors may point. This potential problem is solved by the right-hand rule. We will use the notation  v x w - hence the name "cross product." 
If v = w, then these "two" vectors are one, and there is no single plane determined by them. Thus, we require that v x v = 0.
We define the cross product for pairs of the basic unit vectors i, j, and k. Each of these is perpendicular to the plane of the other two, so we can define i x  j  to be k. The usual convention for coordinates in space is the right-hand rule, as illustrated in the following figure:

That is, if we follow the direction of the fingers to go from the x-axis to the y-axis, then the thumb points in the direction of the z-axis. The unit vectors i, j, and k  point in these same directions. Thus, if we adopt the right-hand rule for cross products as well, then we want i x  j  to be k, as illustrated in the following picture:

Adapted from 

https://www.math.duke.edu/education/ccp/materials/mvcalc/vectors/vec4.html