What are the x- and y-components of vector shown in FIGURE EX3.3 in terms of the angle and the magnitude E?

Verified step by step guidance

1. Identify the vector \( \vec{A} \) and its angle \( \theta \) with respect to the x-axis.
2. Recall that the x-component of a vector can be found using the cosine function: \( A_x = A \cos(\theta) \).
3. Recall that the y-component of a vector can be found using the sine function: \( A_y = A \sin(\theta) \).
4. Substitute the magnitude of the vector \( E \) for \( A \) in the equations: \( A_x = E \cos(\theta) \) and \( A_y = E \sin(\theta) \).
5. Thus, the x-component of the vector is \( E \cos(\theta) \) and the y-component of the vector is \( E \sin(\theta) \).

Key Concepts

Vector Components

Vectors can be broken down into components along the axes of a coordinate system. For a vector at an angle θ, the x-component is found using the cosine function, while the y-component is determined using the sine function. This decomposition allows for easier calculations in physics, as it simplifies the analysis of forces and motion.

Trigonometric Functions

Trigonometric functions, specifically sine and cosine, relate the angles of a triangle to the ratios of its sides. In the context of vector components, the cosine of the angle θ gives the ratio of the adjacent side (x-component) to the hypotenuse (magnitude of the vector), while the sine gives the ratio of the opposite side (y-component) to the hypotenuse. These functions are essential for resolving vectors.

Magnitude of a Vector

The magnitude of a vector represents its length and is a measure of its strength or size. In this case, the vector E is the hypotenuse of a right triangle formed by its x- and y-components. The relationship between the magnitude and its components is given by the Pythagorean theorem, which states that the square of the magnitude equals the sum of the squares of its components.