Lecture Notes: Vectors in Mechanics

Introduction to Vectors

• Definition: A vector has both size and direction.
• In mechanics, vectors are often considered in two dimensions (x and y directions).
• Unit Vectors:
  • i: One unit in the x-direction.
  • j: One unit in the y-direction.

Representing Vectors

• Example: Vector A could be represented as 3i + 2j.
  • 3 units in the x-direction, 2 units in the y-direction.
  • Negative values indicate movement left (for i) or down (for j).

Calculating Magnitudes and Angles

• Magnitude: Use Pythagoras’ theorem.
  • Example: Magnitude of A = ( \sqrt{3^2 + 2^2} = \sqrt{13} )
• Angles:
  • Calculate using trigonometry (tangent function).
  • Example: ( \tan(\theta) = \frac{2}{3} ) gives an angle with the positive x-axis.

Special Cases with Direction

• Handling vectors with directions like (-2i + j).
  • Use positive values for base calculations.
  • Adjust angles based on quadrant location.

Example Calculations

• Find magnitude and angles for vectors like 6i - 7j.
  • Sketch helps visualize directions and angles.

Combining Vectors

• Addition: Add corresponding i and j components.
  • Example: A + B = (3+5)i + (2-4)j = 8i - 2j.
• Subtraction: Subtract corresponding components.
  • Example: 3A - B.

Finding Magnitude of Combined Vectors

• Expand and calculate for combined vectors like 2A + 5B.

Velocity and Speed

• Velocity: A vector quantity with direction.
• Speed: Scalar, just the magnitude of velocity.
• Example: Velocity = 3i + j, Speed = (\sqrt{10}).

Displacement and Distance

• Displacement: Vector from start to end point.
• Distance: Total path length traveled.

Problem Solving with Vectors

• Initial Position and Direction:
  • Given initial position as a vector.
  • Calculate new position using direction and speed over time.

Application of Velocity and Acceleration

• Use formulas like ( v = u + at ) for calculations.
• Example: Given initial and final velocities, find speed.

Force and Acceleration

• Force Equation: ( F = ma ).
  • Calculate acceleration first, then find force.
  • Example: Calculate force when given mass and acceleration vector.

Practice Questions

• Includes exercises on vector addition, magnitude, and velocity.