Grade 11 Physical Sciences: Vectors in 2D Components and Calculations

Decomposition of Vectors in 2D

• Diagonal vectors are complex entities that can be broken down or decomposed into simpler parts known as components.
• These components are typically divided into two specific directions:
  • Horisontal ($x$) component.
  • Vertikale ($y$) component.

Resultant Force and the Triangle of Forces

• The concept of "Kragtedriehoeke" (Triangle of Forces) is utilized to determine the resulting force when multiple forces act in a non-linear fashion.
• This method allows for the graphical or mathematical representation of vectors that are not directed along the same straight line.

Methods for Vector Addition

• Tail-to-Head Method (Stert-aan-kop):

  • This principle is specifically applied to successive or consecutive vectors, such as when following a specific path or route.
  • Positioning: The vectors are arranged so that the arrowhead (the "kop" or head) of the first vector is placed precisely at the back end (the "stert" or tail) of the succeeding vector.
  • The Resultant: The resultant vector is drawn starting from the tail of the very first vector in the sequence and terminating at the head of the final vector.

• Parallelogram Method:

  • This method is used for vectors that act gelyktydig (simultaneously) on the same object or point of application.
  • The Resultant: In this representation, the resultant vector is the hoeklyn (diagonal) of a parallelogram formed by the two vectors. This diagonal must begin at the common origin point where the tails of the original vectors meet.

Systematic Calculation of the Resultant Vector

To find the magnitude and direction of a resultant vector mathematically, follow this four-step procedure:

1. Component Calculation: Calculate the individual horisontal ($x$) and vertikale ($y$) components for every single force acting on the system.
   • Typically, these are found using: $F_x = F \times \tan(x)$ and $F_y = F \times \tan(y)$, or specifically: $F_x = F \times \text{cos}(\theta)$ and $F_y = F \times \text{sin}(\theta)$.
2. Summation: Calculate the net or resultant components for both the $x$ and $y$ axes by summing all individual components ($\text{Resultante x}$ and $\text{Resultante y}$).
3. Magnitude Determination: Apply the Theorem of Pythagoras to the resulting sums to find the magnitude of the resultant vector ($R$):
   • $R^2 = R_x^2 + R_y^2$
   • $R = \text{sqrt}(R_x^2 + R_y^2)$
4. Direction Determination: Calculate the specific angle ($\theta$) of the resultant vector by utilizing the following trigonometric equation:
   • $\tan(\theta) = \frac{R_y}{R_x}$

Rules for Manipulating Vector Triangles

When rearranging or manipulating vector arrows (for example, to form a triangle of forces), certain properties must remain invariant to ensure the vector remains physically identical:

• Length of the Arrow (Grootte): The magnitude must remain the same; you cannot change the scale of the arrow.
• Angle of the Arrow (Rigting): The direction or orientation of the arrow relative to a reference frame must remain constant; the vector cannot be rotated.