The analytical method of the components allows to find the results for the questions about the sum vector are:
a) In the attached we have a scheme of the vectors
b) the modulus is C = 96.3 m
c) The angle is θ = 27.0º
Given parameters
- Vector A with modulus A = 40.0 m and an angle of θ₁ = -20º
- Vector B has a modulus B = 75.0 m and an angle of θ₂ = 50º
To find
a) Draw the vectors and their sum.
b) The module.
c) the adirection.
The sum of vectors has several methods of resolution:
- Graphic. In this case the vectors are drawn and the second is placed on the tip of the first and the resulting vector goes from the origin of the first to the tip of the last. This method is complicated when there are several vectors.
- Analytical. This method is very precise and is the most used when there are many vectors.
The analytical method consists:
- Decompose each vector into its components with respect to a given reference frame, using trigonometry.
- Find the sum of each component
- Construct the final vector using trigonometry and the Pythagorean theorem.
In the attached we have a diagram of each vector and its sum. Let's use trigonometry to find the component of each vectors.
Vector A
cos θ₁ = [tex]\frac{A_x}{A}[/tex]
sin θ₁ = [tex]\frac{A_y}{A}[/tex]
Aₓ = A cos θ₁
[tex]A_y[/tex] = A sin θ₁
Aₓ = 40 cos (-20) = 37.59 m
[tex]A_y[/tex]= 40 sin (-20) = -13.68 m
Vector B
cos θ₂ = [tex]\frac{B_x}{B}[/tex]
sin θ₂ = [tex]\frac{B_y}{B}[/tex]
Bₓ = B cos θ₂
[tex]B_y[/tex] = B sin θ₂
Bₓ = 75 cos 50 = 48.21 m
[tex]B_y[/tex] = 75 sin 50 = 57.45 m
we add the components.
Cₓ = Aₓ + Bₓ
[tex]C_y = A_y + B_y[/tex]
Cₓ = 37.59 + 48.20 = 85.8 m
Cy = -13.68 + 57.45 = 43.8 m
b) We use the Pythagorean theorem to find the modulus of the resultant vector.
C² = Cₓ² + [tex]C_y^2[/tex]
C = [tex]\sqrt{85.8^2 + 43.8^2 }[/tex]
C = 96.3 m.
c) We use trigonometry to find the angle of the resultant vector.
tan θ =[tex]\frac{C_y}{C_x}[/tex]
θ = tan⁻¹ [tex]\frac{C_y}{C_x}[/tex]
θ = tan⁻¹ [tex]\frac{43.8}{85.8}[/tex]
θ = 27.0º
In conclusion, using the analytical method of the components we can find the results for the questions about the sum vector are:
a) In the attached we have a scheme of the vectors
b) the modulus is C = 96.3 m
c) The angle is θ = 27.0º
Learn more about vector addition here: brainly.com/question/25681603
