Scalars and Vectors Study Notes

Scalars and Vectors Study Notes

Scalars

• Definition: A scalar is a physical quantity that has only a magnitude.
• Examples:
  • Mass
  • Length
  • Time
  • Temperature
  • Volume
  • Density

Vectors

• Definition: A vector is a physical quantity that has both a magnitude and a direction.
• Examples:
  • Position
  • Displacement
  • Velocity
  • Acceleration
  • Momentum
  • Force
• Displacement Example: The displacement from the USA to China is 11600 km, in an eastward direction.

Representation of Vectors

• Components:
  • Vectors are typically represented as arrows with:
  • Head: Indicates the direction of the vector.
  • Tail: Indicates the starting point.
• Symbolic Representation: Symbolically represented as $ extbf{AB}$ or with a single capital letter with an arrow above it.

Comparison: Scalar vs Vector

• Scalar Quantity:

  • Has magnitude only.
  • Does not have direction.
  • Specified by a number and a unit (e.g., temperature, speed).

• Vector Quantity:

  • Has both magnitude and direction.
  • Specified by a number along with the direction and unit.
  • Represented by a bold symbol or an arrow sign above its symbol (e.g., acceleration, velocity).

Vector Addition and Subtraction

• Addition of Vectors: If two vectors $ extbf{A}$ and $ extbf{B}$ are represented, their resultant vector, $ extbf{C}$ can be defined as:

  • $extbf{C} = extbf{A} + extbf{B}$

• Subtraction of Vectors: The subtraction of vector $ extbf{B}$ from vector $ extbf{A}$ is defined as the addition of vector $- extbf{B}$ to vector $ extbf{A}$:

  • $extbf{A} - extbf{B} = extbf{A} + (- extbf{B})$

Unit Vectors

• Definition: A unit vector is a vector that has a magnitude of 1 and indicates direction.
• Properties:
  • Lacks both dimension and unit.
  • Serves to specify a direction in space.
• Expression: Any vector $ extbf{A}$ can be expressed as:
  • $extbf{A} = | extbf{A}| extbf{A}_{ ext{unit}}$

Resolution of a Vector

• Definition: The process of breaking a vector into two or more component vectors that can reproduce the original vector’s effect.
• Rectangular Components of 2D Vectors:
  • Vectors can be expressed in component form:
  • $extbf{A} = A<em>{x} extbf{i} + A</em>{y} extbf{j}$
  • Where:
  • $A_x = A imes ext{cos}( heta)$
  • $A_y = A imes ext{sin}( heta)$

Magnitude and Direction from Components

• Calculating Magnitude:
  • $A = ext{sqr}(A<em>x^2 + A</em>y^2)$
• Calculating Direction:
  • Using tangent:
  • heta = an^{-1}igg(rac{Ay}{Ax}igg)
  • alternatively, $A<em>y = A imes ext{sin}( heta), ext{ and } A</em>x = A imes ext{cos}( heta)$

3D Vectors

• Rectangular Components of 3D Vectors:
  • $extbf{A} = A<em>{x} extbf{i} + A</em>{y} extbf{j} + A_{z} extbf{k}$
• Magnitude in 3D:
  • $|A| = ext{sqr}(A<em>x^2 + A</em>y^2 + A_z^2)$
  • Direction angles:
  • eta = ext{cos}^{-1}igg(rac{A_x}{|A|}igg)

Vector Products

1. Multiplying a Vector by a Scalar:

   • If a vector $ extbf{A}$ is multiplied by scalar $s$, the new vector has a magnitude equal to $|A| imes |s|$ and direction dictated by the sign of $s$.

2. Scalar Product (Dot Product): Produces a scalar outcome:

   • extbf{A} ullet extbf{B} = | extbf{A}| | extbf{B}| ext{cos} heta
   • Work Example: Work done, W = extbf{F} ullet extbf{s} = |F| |s| ext{cos} heta

3. Vector Product (Cross Product): Produces a vector:

   • $extbf{A} imes extbf{B} = | extbf{A}| | extbf{B}| ext{sin} heta extbf{n}$

Properties of Scalar and Vector Products

• Scalar Product Properties:

  • The scalar product of any two identical unit vectors equals one (e.g., $ extbf{i} ullet extbf{i} = 1$).
  • The scalar product of different unit vectors equals zero (e.g., $ extbf{i} ullet extbf{j} = 0$).

• Vector Product Properties:

  • The cross product of two identical unit vectors gives zero.
  • The cross product of different unit vectors results in a third unit vector (e.g., $ extbf{i} imes extbf{j} = extbf{k}$).

Exercises

• Exercise examples providing calculations involving vectors, magnitudes, and angles between vectors illustrating practical applications and foundational concepts.