Vectors in Physics

Definition

• A vector is a quantity that has both magnitude and direction.
• Common examples include displacement, velocity, acceleration, and force.

Representation

• Vectors are often represented graphically as arrows:
  • The length of the arrow indicates the magnitude.
  • The direction of the arrow indicates the direction of the vector.
• Algebraically, a vector can be expressed in component form: v = (v₁, v₂) for 2D or v = (v₁, v₂, v₃) for 3D.

Operations

1. Addition

   • Vectors can be added graphically using the tip-to-tail method or mathematically by adding corresponding components.
   • If A = (a₁, a₂) and B = (b₁, b₂), then A + B = (a₁ + b₁, a₂ + b₂).

2. Subtraction

   • To subtract vectors, reverse the direction of the vector being subtracted.
   • A - B = A + (-B).

3. Scalar Multiplication

   • A vector can be multiplied by a scalar, changing its magnitude but not its direction (if the scalar is positive).
   • If k is a scalar and A = (a₁, a₂), then kA = (ka₁, ka₂).

Magnitude

• The magnitude of a vector v = (v₁, v₂) is calculated as:
  • |v| = √(v₁² + v₂²) in 2D.
  • |v| = √(v₁² + v₂² + v₃²) in 3D.

Unit Vectors

• A unit vector is a vector with a magnitude of 1.
• It indicates direction only and can be found by dividing a vector by its magnitude.
• Common unit vectors in 2D:
  • i = (1, 0)
  • j = (0, 1)

Dot Product

• The dot product of two vectors A and B is a scalar and is calculated as:
  • A · B = a₁b₁ + a₂b₂ for 2D vectors.
• The dot product is related to the angle θ between vectors by:
  • A · B = |A||B|cos(θ).

Cross Product

• The cross product is defined only in three dimensions and yields a vector.
• If A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃), then the cross product is:
  • A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁).
• The resultant vector is perpendicular to both A and B.

Applications

• Used in physics to describe forces, motion, and fields.
• Essential in engineering, computer graphics, and navigation.

Vectors

• A vector is a quantity that has both magnitude and direction.
• Examples include displacement, velocity, acceleration, and force.
• Represented graphically as arrows; the length indicates magnitude and the direction the arrow points is the direction of the vector.
• Can be expressed in component form; **v** = (v₁, v₂) for 2D or **v** = (v₁, v₂, v₃) for 3D.

Vector Operations

• Addition can be done graphically using the tip-to-tail method or mathematically by adding corresponding components.
• If **A** = (a₁, a₂) and **B** = (b₁, b₂) then **A** + **B** = (a₁ + b₁, a₂ + b₂)
• Subtraction involves reversing the direction of the vector being subtracted. `A - B = A + (-B).
• Scalar Multiplication involves multiplying a vector by a scalar, which changes its magnitude but not its direction (if the scalar is positive). k**A** = (ka₁, ka₂).

Vector Magnitude

• The magnitude of a vector is calculated by using the Pythagorean theorem:
• |**v**| = √(v₁² + v₂²) in 2D
• |**v**| = √(v₁² + v₂² + v₃²) in 3D.

Unit Vectors

• Unit vectors have a magnitude of 1, indicating direction only, and are found by dividing a vector by its magnitude.
• Common unit vectors in 2D:
  • i = (1, 0)
  • j = (0, 1)

Dot Product

• The dot product is a scalar.
• Calculated as: **A** · **B** = a₁b₁ + a₂b₂ for 2D vectors.
• Measures the "projection" of one vector onto another.
• **A** · **B** = |**A**||**B**|cos(θ).

Cross Product

• Defined only in three dimensions and yields a vector.
• If **A** = (a₁, a₂, a₃) and **B** = (b₁, b₂, b₃) then the cross product:
• **A** × **B** = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁).
• The resultant vector is perpendicular to both A and B.

Vector Applications

• Used in physics to describe forces, motion, and fields.
• Essential in engineering, computer graphics, and navigation.