**a** - According to the reference system, the position vectors (the red ones) are:
** **** ***r*_{D} **= 1 km î ****—** 3 km ĵ
** **** ** r_{S} **= ****—** 0,5 km î **—** 1 km ĵ
** **r_{C} **= 4 km î ****+ ** 1 km ĵ
**î **and **ĵ** are vectors of modulus **1**, also called ``Versors´´, their objetive is to join the vectorial characteristics of a number and tell where it is pointing to. Usually it is portrayed with a little hat or a little comma above ( replacing the typical **i **and **j** ) . Versor **î** has the same direcction and sense in the **x** axis, as the **ĵ** versor in the **y** axis.
**b - **The displacement vector, **Δr**_{DS }(the bus from Diego´s house to Silvia´s) is the vectorial subtraction between the rear position vector and it´s previous one, **r**_{S} **—** **r**_{D}; To find it, you must subtract component by component. It´s easy, just look.
**Δr**_{DS }= ** **r_{S} **—** **r**_{D}
**Δr**_{DS}** **= **— **1,5 km î **+ **2 km **ĵ**
This is the green one, with it´s origin in Diego´s house and it´s end in Silvia´s house, it does not mean that the bus went from one point to the other moving on a straight line as it was an helicopter. It´s about **displacement**, **not trajectory.**
Those displacement vectors are independent from the reference system you have chosen. If you want you can try with other reference system, you will get the same vector.
**Δr**_{SC }= ** **r_{C} **—** **r**_{S}
**Δr**_{SC}** **= 4,5 km î **+ **2 km ĵ
Esos vectores desplazamiento son independientes del **SR** elegido. Probá vos con otro **SR**... vas a ver que te dan lo mismo.
**c** - Las velocidades medias son los cocientes entre los desplazamientos y los intervalos de tiempo insumidos en cada desplazamiento. Eso es lo que nos va a permitir saber a qué hora salió Diego de su casa.
** v**_{mDS} = Δr_{DS} / Δt_{DS}
That average speed is a fact that you have in the statement above... then:
**— **6 km î **+ **8 km ĵ = **(— **1,5 km î **+ **2 km ĵ ) / Δt_{DS}
The division is also applied component by component, so...
**— **6 km î **=**** — **1,5 km î** / Δt**_{DS}
*8 km* ĵ = *2 km* ĵ / Δt_{DS}
Logically, both calculations should give the same answer... and they do: the time spent in the bus is** 0.25 h**, that is **15 minutes**.
That means: Diego left his house at **18:55**. The movie started at **19:30**... but they arrived at **19:35** (5 minutes late). They left Silvia´s house fifteen minutes before the arrival, that is **19:20**. But Diego arrived at Silvia´s house 10 minutes earlies, at **19:10**, and he waited will she was on make up. Women! if he took 15minutes to arrive at her house, that means he left his house at **18:55**.
**d - **to calculate Diego´s average speed vector from his house to the cinema, fisrt we must finf his displacement... (wich I didn´t represent in the scheme) and after that, we divide it by the total time interval:
**Δr**_{DC }= ** **r_{C} **—** **r**_{D}
**Δr**_{DC}** **= 3 km î **+ **4 km ĵ
The time interval between Diego´s departure and his arrival to the cinema is equal to 40 minutes, that is **0,66 h**
**v**_{mDC }= Δr_{DS} / Δt_{DS}
**v**_{mDC }= *( 3 km* î **+ **4 km ĵ ) / 0,66 h
**v**_{mDC }= *4,5 km**/**h* î **+ **6 km*/**h* ĵ
To find the modulus of that vector, we apply Pitagoras:
**|v**_{mDC}| = [ *(**4,5*km*/**h*)*²* **+** (6 km*/**h* *)²* ]^{½}
**|v**_{mDC}| = **7,5** *km**/**h*
(not bad for a taxi) |