Motion in a Plane: Introduction to Vectors

Overview of Motion in One Dimension

• Motion previously studied was in a straight line (one dimension).
• Direction specified by positive and negative signs.
• Simple representation with only two possible directions.

Introduction to Motion in a Plane

• Objects can move in multiple directions in a plane.
• A single positive/negative sign is insufficient to represent direction.
• Introduction of vectors to handle direction in two and three dimensions.

Scalars vs. Vectors

• Scalar Quantities:

  • Have magnitude only (no direction).
  • Examples:
    • Distance (e.g., 5 meters)
    • Time (e.g., 1 hour)
    • Mass (e.g., 10 kg)
    • Volume, Area, Speed.

• Vector Quantities:

  • Have both magnitude and direction.
  • Examples:
    • Displacement (5 meters in a specified direction)
    • Velocity
    • Acceleration
    • Force (mass x acceleration).

Understanding Vectors

• A vector is represented graphically by a directed line segment.
• Position Vector: Represents the position of a point in reference to an origin.
• Displacement Vector: Difference between final and initial position vectors.
• Notation: Vectors represented with an arrow above the letter (e.g., ( \vec{R} \)).

Representation of Vectors

• Magnitude: The length of the vector, represented without direction.
• Equality of Vectors: Two vectors are equal if they have the same magnitude and direction.
• Vectors can be shifted parallel to determine equality.

Operations with Vectors

Multiplying Vectors by Real Numbers

• Multiplying a vector by a positive number scales the magnitude without changing direction.
• Multiplying by a negative number reverses the direction of the vector.
• Multiplying any vector by 0 results in a null vector.

Vector Addition and Subtraction (Graphical Method)

• Triangle Law of Vector Addition: To add vectors, place them head-to-tail.
  • Resultant vector is drawn from the tail of the first to the head of the last vector.
• Parallelogram Law of Vector Addition: If two vectors are represented as adjacent sides of a parallelogram, the resultant is the diagonal.
• Subtraction of Vectors: To subtract, add the negative of the vector.

Properties of Vector Addition

• Commutative Law: ( \vec{A} + \vec{B} = \vec{B} + \vec{A} \)
• Associative Law: ( \vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C} \)

Example Problems

• Example problems in the textbook involve practical scenarios (e.g., motion of a boat across a river).
• Importance of understanding vectors for resolving real-life motion problems.

Conclusion

• Importance of mastering vectors for further study in motion in a plane.
• Next class will cover resolution of vectors and additional concepts.