Question

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)

Answer 1

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= $\frac{2}{3}$(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

$\frac{2}{3}$(A-C): $\frac{2}{3}$ * { [(8.48)i + (8.48)j] - [(-6.12)i - (5.14)j] } = $\frac{2}{3}$ * { (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