Comprehensive Study Guide on Vector Analysis and Physics Applications

Introduction to Vector Quantities

• Vector Representation:
  • Vector quantities are characterized by both magnitude and direction.
  • If the magnitude of a vector is known, it must be written in a specific format to be usable in mathematical formulas and physical equations.

Unit Vectors in Cartesian Systems

• Definition and Properties:

  • A unit vector has a magnitude equal to $1$ (using the unit of the specific quantity of interest).
  • In the Cartesian coordinate system, the unit vectors are denoted as $\hat{i}$ and $\hat{j}$.
  • These unit vectors are treated as constants.
  • Directionality: $\hat{i}$ is always parallel to the x-axis, and $\hat{j}$ is always parallel to the y-axis.

• Vectors Parallel to an Axis:

  • Example: A car moves along the x-axis toward the right.
  • Given magnitude of velocity: $v = 5\,m/s^{-1}$.
  • Vector notation for this velocity: $\mathbf{v} = 5\hat{i}\,m/s^{-1}$.

Drawing and Scaling Vectors

• Vector Components:

  • A force vector $\mathbf{F}$ can be represented as the sum of its horizontal and vertical components.
  • Example: For a vector with an x-component of $3$ and a y-component of $2$, the force is written as: $\mathbf{F} = (3\hat{i} + 2\hat{j})\,N$.

• Proportionality and Magnitude:

  • The drawn length of a vector is directly proportional to its magnitude.
  • Example comparison:
    • If $\mathbf{F}_1 = 2\,N$ and $\mathbf{F}_2 = 1\,N$, then the length of the green vector representing $\mathbf{F}_2$ is half the length of the red vector representing $\mathbf{F}_1$.

Vector Addition in Two Dimensions

• Example Calculation:
  • Given forces:
    • $\mathbf{F}_1 = (2\hat{i} + \hat{j})\,N$
    • $\mathbf{F}_2 = (2\hat{i})\,N$
  • To find the total force (Sum): $\mathbf{F}_{tot} = \mathbf{F}_1 + \mathbf{F}_2$
  • Calculation: $(2\hat{i} + \hat{j} + 2\hat{i})\,N = (4\hat{i} + \hat{j})\,N$
  • Correction based on slide 7 illustration notes: If $\mathbf{F}_1$ is drawn as $\hat{i} + \hat{j}$ and $\mathbf{F}_2$ as $2\hat{i}$, then the result is $(3\hat{i} + \hat{j})\,N$.

Analytical Properties of 2D Vectors

• Magnitude (Length):

  • For a vector $\mathbf{a} = a_x \hat{i} + a_y \hat{j}$, the magnitude is calculated as:
  • $a = |\mathbf{a}| = \sqrt{a_x^2 + a_y^2}$

• Scalar (Dot) Product:

  • For two vectors $\mathbf{a} = a_x \hat{i} + a_y \hat{j}$ and $\mathbf{b} = b_x \hat{i} + b_y \hat{j}$:
  • $\mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y$

Exercise 1: Drafting Vectors from Geometry

• Scenario: An equilateral triangle with side length $a = 2\,m$. The side $B\Gamma$ is parallel to the x-axis.
• Part A: Writing vector $BA$ in terms of $\hat{i}$ and $\hat{j}$:
  • The angle in an equilateral triangle is $60^{\circ}$.
  • X-component: $a \times \cos(60^{\circ}) = 2 \times 0.5 = 1\,m$.
  • Y-component: $a \times \sin(60^{\circ}) = 2 \times \frac{\sqrt{3}}{2} \approx 1.73\,m$.
  • Result: $\mathbf{BA} = (1\hat{i} + 1.73\hat{j})\,m$.
• Part B: Finding the unit vector parallel to $BA$:
  • A unit vector $\hat{BA}$ has a magnitude of $1$ and no units.
  • It is calculated by dividing the vector by its magnitude:
  • $\hat{BA} = \frac{\mathbf{BA}}{|\mathbf{BA}|} = \frac{(1\hat{i} + 1.73\hat{j})\,m}{2\,m}$
  • Result: $\hat{BA} = (0.5\hat{i} + 0.87\hat{j})$.

Exercise 2: Position and Displacement

• Scenario: Triangle vertices $A(0,0)$, $B(1,0)$, and $\Gamma(2,1)$, with coordinates in $cm$.
• Finding the vector $B\Gamma$:
  • Calculation method: $B\Gamma = \text{End Point} - \text{Starting Point}$.
  • Calculation: $\mathbf{B\Gamma} = \Gamma - B = (2, 1) - (1, 0) = (1, 1)$.
  • Vector format: $\mathbf{B\Gamma} = (\hat{i} + \hat{j})\,cm$.

Vectors in Three Dimensions (3D)

• The 3D Coordinate System:

  • Includes three axes: $x$, $y$, and $z$.
  • Uses a right-handed system (described by the "Rechte-hand-regel").
  • Locating a point $(x, y, z)$ (e.g., $A(1, 2, 1)$) involves three steps moving parallel to each respective axis.

• Orientation relative to a 2D plane:

  • Vectors can be perpendicular to the paper/screen.
  • Direction from the paper/screen toward the viewer (represented visually by the top of a screw).
  • Direction away from the viewer and into the paper/screen (represented by the back of a screw).

• 3D Vector Notation:

  • A vector in 3D is written as: $\mathbf{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$.

Practical 3D Application: Hiking Displacement

• Scenario: Hiking from a University to the summit of Mitsikeli.
• Data Points:
  • University Altitude: $530\,m$.
  • Mitsikeli Summit Altitude: $1810\,m$.
  • Horizontal displacement (x): $\Delta x = 6\,km$.
  • Horizontal displacement (y): $\Delta y = 10.5\,km$.
  • Vertical displacement (z): $\Delta z = 1810\,m - 530\,m = 1280\,m = 1.28\,km$.
• Displacement Vector:
  • $\Delta \mathbf{r} = (6\hat{i} + 10.5\hat{j} + 1.28\hat{k})\,km$.

Mathematical Operations with Vectors

• Addition and Subtraction:

  • Performed by adding or subtracting the corresponding components of the vectors.
  • $(a_x\hat{i} + a_y\hat{j} + a_z\hat{k}) \pm (b_x\hat{i} + b_y\hat{j} + b_z\hat{k}) = (a_x \pm b_x)\hat{i} + (a_y \pm b_y)\hat{j} + (a_z \pm b_z)\hat{k}$.

• The Resultant Vector ($\mathbf{F}_{ol}$ or $\mathbf{F}_{total}$):

  • The resultant is the algebraic sum of all component forces.
  • The sign of a component indicates its direction and is not always positive.
  • Example: $\mathbf{F}_1 = (\hat{i} + 3\hat{j})$ and $\mathbf{F}_2 = (2\hat{i} + \hat{j})$ results in $\mathbf{F}_{ol} = (3\hat{i} + 4\hat{j})$.

• Interior (Dot) Product:

  • Geometric Definition: A scalar value defined by $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\phi)$, where $\phi$ is the angle between the vectors.
  • Algebraic Definition: Equivalent result derived from unit vector properties:
    • $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$
    • $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{i} \cdot \hat{k} = 0$
  • Calculation: $\mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y + a_z b_z$.
  • Example with the forces above: $\mathbf{F}_1 \cdot \mathbf{F}_2 = (1 \times 2) + (3 \times 1) = 2 + 3 = 5$.

Vector Magnitude and Normalization

• Calculating Magnitude in 3D:

  • Derived from the dot product with itself: $a^2 = \mathbf{a} \cdot \mathbf{a}$.
  • $a = \sqrt{a_x^2 + a_y^2 + a_z^2}$.
  • Example for $\mathbf{F}_1 = \hat{i} + 3\hat{j}$: $|\mathbf{F}_1| = \sqrt{1^2 + 3^2} = \sqrt{10} \approx 3.16$.

• Constructing Unit Vectors:

  • To find a unit vector $\hat{a}$ in the same direction as $\mathbf{a}$, divide the vector components by the magnitude: $\hat{a} = \frac{a_x}{a}\hat{i} + \frac{a_y}{a}\hat{j} + \frac{a_z}{a}\hat{k}$.
  • If a unit vector is multiplied by a scalar $x$, the magnitude of the new vector becomes $x$.

Computational Exercise: Bond Angle Calculation

• Target: Calculate the H-C-H bond angle in a molecule.
• Given Coordinates (in Angstroms [\u00c5]):
  • Vector $\mathbf{a} = -0.7145\hat{i} - 0.4125\hat{j}$
  • Vector $\mathbf{b} = +0.7145\hat{i} - 0.4125\hat{j}$
• Magnitudes:
  • $|\mathbf{a}| = 0.825$
  • $|\mathbf{b}| = 0.825$
• Calculating the Angle (\u03c6):
  • $\cos(\phi) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$
  • Dot product: $(-0.7145) \times (0.7145) + (-0.4125) \times (-0.4125) = -0.5105 + 0.1701 = -0.34$
  • $\cos(\phi) = \frac{-0.34}{0.825 \times 0.825} = -0.4995$
  • Solving for $\phi$: The common tetrahedral-related value often results in $\theta \approx 120^{\circ}$ in planar models or $114^{\circ}$ based on specific molecular constraints shown.

Applications in Physics and Geometry

• Common Vector Quantities:

  • Position (Θέση)
  • Displacement (Μετατόπιση)
  • Velocity (Ταχύτητα)
  • Acceleration (Επιτάχυνση)
  • Force ($\mathbf{F}$, Δύναμη)

• Analytical vs. Euclidean Geometry:

  • Analytical geometry is significantly more practical for physics than Euclidean geometry.
  • It allows physical laws to be written in concise vector form.
  • Enables algebraic solutions to complex problems and integration with calculus (differential and integral analysis).