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draw-unit-vectors-in-a-plane

🚀 Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. In kinematics, vectors are used to represent quantities that have both magnitude and direction. When analyzing motion in a plane, it is often necessary to resolve vectors into their components along the coordinate axes. This process involves drawing unit vectors, which are vectors with a magnitude of one, in the direction of the axes. Unit vectors are typically denoted as \( \hat{i} \) for the x-axis and \( \hat{j} \) for the y-axis. By using these unit vectors, any vector in the plane can be expressed as a combination of its components along the x and y directions.

Theory Explanation

Understanding Vectors

A vector is a quantity that has both magnitude and direction. In a two-dimensional plane, a vector can be represented as an arrow pointing from one point to another. The length of the arrow represents the magnitude, while the direction of the arrow indicates the direction of the vector.

Unit Vectors

A unit vector is a vector that has a magnitude of one. In a Cartesian coordinate system, the unit vectors are defined as \( \hat{i} \) and \( \hat{j} \), which point in the positive x and y directions, respectively. Any vector \( \vec{A} \) can be expressed in terms of its components along these unit vectors.

Resolving Vectors into Components

To resolve a vector into its components, you can use trigonometric functions. For a vector \( \vec{A} \) with an angle \( \theta \) with respect to the x-axis, the components can be calculated as follows: \( A_x = A \cos(\theta) \) and \( A_y = A \sin(\theta) \). This allows you to express the vector as \( \vec{A} = A_x \hat{i} + A_y \hat{j} \).

Drawing Unit Vectors

To draw unit vectors in a plane, start by drawing the x and y axes. Then, draw an arrow of length 1 unit in the direction of the x-axis to represent \( \hat{i} \) and another arrow of length 1 unit in the direction of the y-axis to represent \( \hat{j} \). These unit vectors can be used to represent any vector in the plane.

Key Points

  • 🎯 Vectors have both magnitude and direction.
  • 🎯 Unit vectors have a magnitude of one and are used to represent directions.
  • 🎯 Vectors can be resolved into components using trigonometric functions.
  • 🎯 The components of a vector can be expressed in terms of unit vectors.
  • 🎯 Drawing unit vectors helps visualize vector addition and resolution.

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Examples:💡

Resolve the vector \( \vec{A} \) with a magnitude of 5 units at an angle of 30 degrees to the x-axis into its components.

Solution:

Step 1: Calculate the x-component using \( A_x = A \cos(\theta) \). Here, \( A = 5 \) and \( \theta = 30^\circ \). Thus, \( A_x = 5 \cos(30^\circ) = 5 \times \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2} \).

Step 2: Calculate the y-component using \( A_y = A \sin(\theta) \). Thus, \( A_y = 5 \sin(30^\circ) = 5 \times \frac{1}{2} = 2.5 \).

Step 3: Express the vector in terms of unit vectors: \( \vec{A} = A_x \hat{i} + A_y \hat{j} = \frac{5\sqrt{3}}{2} \hat{i} + 2.5 \hat{j} \).

Draw the unit vectors \( \hat{i} \) and \( \hat{j} \) in a Cartesian plane.

Solution:

Step 1: Draw the x-axis and y-axis on a graph.

Step 2: Draw an arrow of length 1 unit along the x-axis to represent \( \hat{i} \).

Step 3: Draw an arrow of length 1 unit along the y-axis to represent \( \hat{j} \).

Common Mistakes

  • Mistake: Confusing the direction of the components; students may mix up \( A_x \) and \( A_y \).

    Correction: Always refer to the angle with respect to the x-axis when calculating components.

  • Mistake: Forgetting to use the correct trigonometric functions when resolving vectors.

    Correction: Remember that \( A_x = A \cos(\theta) \) and \( A_y = A \sin(\theta) \).

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