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resolve-vector-into-rectangular-components

🚀 Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. One important aspect of kinematics is the resolution of vectors into their rectangular components. This process allows us to break down a vector into its horizontal and vertical components, making it easier to analyze motion in two dimensions. For example, if a car is moving at an angle to the ground, we can resolve its velocity vector into horizontal and vertical components to understand its motion better.

Theory Explanation

Understanding Vectors

A vector is a quantity that has both magnitude and direction. In kinematics, vectors are used to represent quantities such as displacement, velocity, and acceleration. To resolve a vector, we need to understand its direction and how it can be represented in a coordinate system.

Rectangular Components

The rectangular components of a vector are the projections of the vector along the axes of a coordinate system, typically the x-axis (horizontal) and y-axis (vertical). For a vector \( \vec{A} \) at an angle \( \theta \), the components can be found using trigonometric functions: \( A_x = A \cos(\theta) \) and \( A_y = A \sin(\theta) \).

\[ A_x = A \cos(\theta), A_y = A \sin(\theta) \]
Resolving a Vector

To resolve a vector into its components, follow these steps: 1. Identify the magnitude of the vector and the angle it makes with the horizontal axis. 2. Use the trigonometric functions to calculate the horizontal and vertical components. 3. Label the components appropriately as \( A_x \) and \( A_y \).

Key Points

  • 🎯 Vectors have both magnitude and direction.
  • 🎯 The components of a vector can be found using trigonometric functions.
  • 🎯 Resolving vectors simplifies the analysis of motion in two dimensions.

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Examples:💡

Example 1: Resolve a vector of 10 m at an angle of 30° to the horizontal.

Solution:

Step 1: Identify the magnitude of the vector (10 m) and the angle (30°).

Step 2: Calculate the horizontal component: \( A_x = 10 \cos(30°) = 10 \times \frac{\sqrt{3}}{2} \approx 8.66 \, m \).

\[ A_x = 10 \cos(30°) = 10 \times \frac{\sqrt{3}}{2} \approx 8.66 \]

Step 3: Calculate the vertical component: \( A_y = 10 \sin(30°) = 10 \times \frac{1}{2} = 5 \, m \).

\[ A_y = 10 \sin(30°) = 10 \times \frac{1}{2} = 5 \]

Example 2: Resolve a vector of 15 m at an angle of 45° to the horizontal.

Solution:

Step 1: Identify the magnitude of the vector (15 m) and the angle (45°).

Step 2: Calculate the horizontal component: \( A_x = 15 \cos(45°) = 15 \times \frac{\sqrt{2}}{2} \approx 10.61 \, m \).

\[ A_x = 15 \cos(45°) = 15 \times \frac{\sqrt{2}}{2} \approx 10.61 \]

Step 3: Calculate the vertical component: \( A_y = 15 \sin(45°) = 15 \times \frac{\sqrt{2}}{2} \approx 10.61 \, m \).

\[ A_y = 15 \sin(45°) = 15 \times \frac{\sqrt{2}}{2} \approx 10.61 \]

Common Mistakes

  • Mistake: Confusing the angle used for sine and cosine functions.

    Correction: Always remember that cosine is used for the horizontal component and sine for the vertical component.

  • Mistake: Forgetting to convert angles to radians when using a calculator set to radian mode.

    Correction: Ensure your calculator is set to the correct mode (degrees or radians) based on the angle measurement.

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