## 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

## 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:

- X Component (D
_{x}): 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. D_{x}= D * cos(α), D_{y}= 3 m * cos(45°) - Y Component (D
_{y}): 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. D_{y}= D_{y}* sin(α), D_{y}= 3 m * sin(45°)

Calculating the values:

D_{x} ≈ 2.12 m

D_{y} ≈ 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:

- X Component (E
_{x}): Calculate the horizontal component using the cosine of the angle β. E_{x}= E * cos(β) Ex = 4.5 m * cos(37°) - Y Component (E
_{y}): Calculate the vertical component using the sine of the angle β. E_{y}= E * sin(β), E_{y}= 4.5 m * sin(37°)

Calculating the values:

E_{x} ≈ 3.6 m

E_{y} ≈ 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.