î and ĵ are vectors of modulus 1, called versores, 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_{AB}, is the vectorial substraction between the position vectors (you take the final one and subtract the one from the beginning), r_{B} — r_{A}; to find it, you have to subtract component by component Just look.
Δr_{AB }= r_{B} — r_{A}
Δr_{AB} = (90 km — 160 km ) î + ( —120 km — 120 km) ĵ
Δr_{AB} = —70 km î — 240 km ĵ
I portrayed it in green. You will find them repeated twice; The thing above is the geometric operation: to subtract two vectors you just have to connect both ends. The subtract vector always has it´s origin in the first one and ends in the second one.
If I portray the analytically obtained subtract vector in the same reference system as the position vectors, I get the exact same vector, but centered in the origin of the reference system. Give it a good look, you have it there.
c -The average speed
v_{m} = Δr / Δt
To analytically obtain the vector v_{m}, you hae to divide each component of the Δr by the time interval. Look:
v_{m} = — 35 km/h î — 120 km/h ĵ
The modulus of a vector (how lenghty is it´s portrayal) is obtained by the theorem of Pitagoras.
|v_{m}| = [ ( —35 km/h)² + (120 km/h )² ]^{½}
|v_{m}| = 125 km/h
d - This... This I´ll leave it to you. I´ll just leave you a guide and the answers.
Δt_{BM } = 1,5 h, Δt_{MA } = 1,66 h
v_{m}_{BM} = — 60 km/h î + 80 km/h ĵ
v_{m}_{MA} = 96 km/h î + 72 km/h ĵ
v_{m}_{AA} = 0 km/h î + 0 km/h ĵ
e -You draw the other two trajectories. If you are going to draw on my drawing you are best by printing it first. Because drawing on the glass of the monitor is incovenient. Realice: there is an infinite amount of possible trajectories, it is not necessary for the plane to travel rectilinearly, following the vectors, The displacement depends exclusively on the position of the final and beginning of the trip. That´s the definition, do not get angry about that, It´s usefull and that´s all. The path of the plane, the length of the total travel, call it as you want... is not the same as displacement (at least in the physics language). |