Utilizando los siguientes vectores, realice las operaciones
que se indican:
|A|=12cm, θ=45º ; |B|=17cm, θ=120º ; |C|=8cm, θ=220º ;
|D|=5cm, θ=300º
a) R1=A+2B-(D-C)
b) R2= 2/3(A-C)+4(B+D)
Respuestas
Las operaciones que se indican son: R1 = (-17.14)i + (37.11)j y R2 = (-14.27)i + (50.64)j
Por definición, las coordenadas x e y de un vector se calculan con las siguientes expresiones:
x: |A| * cos(θ)
y: |A| * sen(θ)
El vector A sería:
A = [|A| * cos(θ)]i + [|A| * sen(θ)]j
A = [12 * cos(45°)]i + [12 * sen(45°)]j
A = (8.48)i + (8.48)j
B = [|B| * cos(θ)]i + [|B| * sen(θ)]j
B = [17 * cos(120°)]i + [17 * sen(120°)]j
B = (-8.5)i + (14.72)j
C = [|C| * cos(θ)]i + [|C| * sen(θ)]j
C = [8 * cos(220°)]i + [8 * sen(220°)]j
C = (-6.12)i - (5.14)j
D = [|D| * cos(θ)]i + [|D| * sen(θ)]j
D = [5 * cos(300°)]i + [5 * sen(300°)]j
D = (2.5)i - (4.33)j
Teniendo el valor de los vectores podemos calcular las operaciones solicitadas:
a)R1=A+2B-(D-C)
Calculamos los vectores resultantes:
2B: 2 * [(-8.5)i + (14.72)j] = (-17)i + (29.44)j
-(D-C): - { [(2.5)i - (4.33)j] - [(-6.12)i - (5.14)j] } = - {(8.62)i + (0.81)j} = (-8.62)i - (0.81)j
R1 = [(8.48)i + (8.48)j] + [(-17)i + (29.44)j] + [(-8.62)i - (0.81)j]
R1 = (-17.14)i + (37.11)j
b)R2= (A-C)+4(B+D)
Calculamos los vectores resultantes:
4(B+D): 4 * { [(-8.5)i + (14.72)j] + [(2.5)i - (4.33)j] } = 4 * { (-6)i + (10.39)j } = (-24)i + (41.56)j
(A-C):
* { [(8.48)i + (8.48)j] - [(-6.12)i - (5.14)j] } =
* { (14.6)i + (13.62)j } = (9.73)i +(9.08)j
R2 = [(9.73)i +(9.08)j] + [(-24)i + (41.56)j]
R2 = (-14.27)i + (50.64)j