Question

What are the X and Y components of vector Ď. The magnitude of the vector is D=3m, and the angle α=45 degrees b) What are the X- and Y-axis components of vector Ē, the magnitude of the vector is E = 4.5m, and the angle β=37 degree

Answer

a) X-component of Ď: 3m * cos(-45°) = 2.12m
Y-component of Ď: 3m * sin(-45°) = -2.12m

b) X-component of Ē: 4.5m * cos(37°) ≈ 3.57m
Y-component of Ē: 4.5m * sin(37°) ≈ 2.7m

What are the X and Y components of vector Ď
What are the X and Y components of vector Ď

Explanation

Our task is to find the X and Y components of two vectors, Ď and Ē, given their magnitudes and angles. Vector components are essential in physics and engineering to understand how forces or displacements act in different directions.

Vector Ď:

a) To find the X and Y components of vector Ď, which has a magnitude D = 3 meters and an angle α = 45 degrees, we can use trigonometric principles. Here’s how we break it down:

  1. X Component (Dx): This is the component of the vector in the horizontal direction (parallel to the X-axis). We can calculate it using the cosine of the angle α. Dx = D * cos(α), Dy = 3 m * cos(45°)
  2. Y Component (Dy): This is the component of the vector in the vertical direction (parallel to the Y-axis). We can calculate it using the sine of the angle α. Dy = Dy * sin(α), Dy = 3 m * sin(45°)

Calculating the values:

Dx ≈ 2.12 m
Dy ≈ 2.12 m

Vector Ē:

b) To find the X and Y components of vector Ē, which has a magnitude E = 4.5 meters and an angle β = 37 degrees, we will use the same trigonometric principles:

  1. X Component (Ex): Calculate the horizontal component using the cosine of the angle β. Ex = E * cos(β) Ex = 4.5 m * cos(37°)
  2. Y Component (Ey): Calculate the vertical component using the sine of the angle β. Ey = E * sin(β), Ey = 4.5 m * sin(37°)

Calculating the values:

Ex ≈ 3.6 m
Ey ≈ 2.7 m

Summary:

To summarize, we have now found the X and Y components of both vectors:

Vector Ď:
Dx ≈ 2.12 m (X-component)
Dy ≈ 2.12 m (Y-component)

Vector Ē:
Ex ≈ 3.6 m (X-component)
Ey ≈ 2.7 m (Y-component)

These components describe how the vectors Ď and Ē act in the X and Y directions, respectively. Understanding vector components helps us to analyze and solve complex problems involving forces, displacements, and more, which is especially useful when working with objects moving in two-dimensional space.