The analytical method for the sum of vectors allows to find the result for the sum of the vectors is:
-
The resulting vector has a modulus of R = 239.99 m and an angle of θ= 212.8º
The displacement is a vector quantity, therefore vector algebra must be used to perform the addition of vectors, in general there are two methods:
- Graphic. In this case, the origin of a vector is placed at the tip of the previous one and the resulting vector goes from the origin of the first vector to the tip of the last, this method is not very precise.
- Analytical. In this method the vectors are decomposed into a coordinate system, the sum of the components is performed and the resulting vector is built, this method is very expensive.
Indicate in the graph a Cartesian coordinate system, let alone the West - East axis coincide with the x axis. Let's use trigonometry to decompose the vectors.
Vector A.
Aₓ = 100.0 m
Vector B.
B_y = -300.0 m
The negative sign indicates that it goes in the negative direction of the y-axis
Vector C.
We use trigonometry.
The angle measured from the positive side of the x-axis counterclockwise is
θ = 180 + 30 = 210
sin 210 = [tex]\frac{C_y}{C}[/tex]
cos 210 = [tex]\frac{C_x}{C}[/tex]
C_y = C sin 210
Cₓ = C cos 210
C_y = 150.0 sin 210 = -75.0 m
Cₓ = 150.0 cos 210 = -129.9 m
Vector D.
The angle from the positive side of the x-axis counterclockwise.
θ = 180-60 = 120º
sin 120 = [tex]\frac{D_y}{D}[/tex]
cos 120 = [tex]\frac{D_x}{D}[/tex]
D_y = D sin 120
Dₓ = D cos 120
D_y = 200.0 sin 120 = 173.2 m
Dₓ = 200.0 cos 120 = -100.0 m
we add the component with algebraic sum.
x = Aₓ + Cₓ + Dₓ
y = B_y + C_y + D_y
x = 100 - 129.9 -100 = -129.9 m
y = -300 - 75.0 + 173.2 = -201.8 m
We construct the resulting vector.
We use the Pythagorean theorem for the Modulus.
[tex]R+ \sqrt{x^2 +y^2}[/tex]
R = [tex]\sqrt{129.9^2 + 201.8^2 }[/tex]
R = 239.99 m
We use trigonometry for the angle.
tan θ = [tex]\frac{y}{x}[/tex]
θ = [tex]tan^{-1} \frac{y}{x}[/tex]
θ = [tex]tan^{-1} \frac{201.8}{129.9}[/tex]tan-1 (201.8 / 129.9)
θ = 57.2º
Since the two coordinates are negative, this angle is in the third quadrant; to measure it from the positive side of the x-axis.
θ = 270 - θ'
θ = 270 - 57.2
θ = 212.8º
In conclusion using the analytical method for the sum of vectors we can find the result for the sum of the vectors is:
-
The resulting vector has a modulus of R = 239.99 m and an angle of θ= 212.8º
Learn more about vector addition here: brainly.com/question/24855749
