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

Verified step by step guidance

1. Step 1: Identify the components of the vector. The vector shown in the figure has a magnitude E and is oriented at an angle θ with respect to the positive x-axis. The x-component and y-component of the vector can be determined using trigonometric functions.
2. Step 2: Use the cosine function to find the x-component. The x-component of the vector is given by the formula: $Ecos\theta$. This is because the cosine function relates the adjacent side (x-component) to the hypotenuse (magnitude of the vector).
3. Step 3: Use the sine function to find the y-component. The y-component of the vector is given by the formula: $Esin\theta$. This is because the sine function relates the opposite side (y-component) to the hypotenuse (magnitude of the vector).
4. Step 4: Write the vector components in terms of E and θ. The x-component is $Ecos\theta$, and the y-component is $Esin\theta$. These components represent the projection of the vector along the x-axis and y-axis respectively.
5. Step 5: Verify the components using the Pythagorean theorem. The magnitude of the vector E should satisfy the equation: $\sqrt{E^2}$, which is consistent with the sum of the squares of the x-component and y-component.

Key Concepts

Vector Components

Vectors can be broken down into components along the axes of a coordinate system. For a vector A at an angle θ, the x-component (Ax) is found using the cosine function, while the y-component (Ay) is determined using the sine function. This decomposition allows for easier analysis of vector behavior in two-dimensional space.

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, cosine(θ) gives the ratio of the adjacent side (x-component) to the hypotenuse (magnitude of the vector), while sine(θ) gives the ratio of the opposite side (y-component) to the hypotenuse. These functions are essential for calculating the components of vectors.

Magnitude of a Vector

The magnitude of a vector is its length and represents the overall strength or size of the vector. It is denoted as E in this context. The magnitude is crucial for determining the components of the vector, as the x- and y-components are derived from the magnitude multiplied by the respective trigonometric functions of the angle θ.