Vectors: Summation and Components

Vectors

• The moment of a force measures the tendency to cause rotation around a point.

Sum of Vectors

Graphic Method

• Vectors are placed sequentially, maintaining their magnitude, direction, and sense.
• The resultant vector goes from the origin of the first vector to the end of the last vector.
• $$\vec{R} = \vec{A} + \vec{B}$$

Analytical Method

• Concurrent vectors intersect at a single point.
• Vectors are decomposed into rectangular components along x and y axes.
  • (A_x = A \cos\theta)
  • (A_y = A \sin\theta)
• Components are summed along each axis:
  • (R_x = A_x + B_x)
  • (R_y = A_y + B_y)
• The resultant vector's magnitude and direction are calculated:
  • (R = \sqrt{R_x^2 + R_y^2})
  • (\theta = \arctan\left(\frac{R_y}{R_x}\right))

System of Linear Equations: Compatibility

Theory

Definition

• A system of linear equations (SEL) consists of m equations with n unknowns.
  • $a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = b_1$
  • $a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = b_2$
  • ...
  • $a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n = b_m$
• $x_1, x_2, \dots, x_n$ are unknowns.
• $a_{ij} \in \mathbb{R}$ are the coefficients, $i = \overline{1, m}, j = \overline{1, n}$.
• $b_i \in \mathbb{R}$ are the constants, $i = \overline{1, m}$.

Notations

Matrix Form

• $AX = B$
  • $A$ is the system's matrix.
  • $X$ is the matrix of unknowns.
  • $B$ is the matrix of constants.

Augmented Matrix

• $\overline{A} = (A \mid B)$ includes both the coefficients and the constants.

Definitions

• A solution is an n-tuple $(s_1, s_2, \dots, s_n) \in \mathbb{R}^n$ that satisfies all equations.
• Solving a system means finding all solutions. The set of all solutions is denoted by $S$.

Compatibility of Linear Equation Systems

Definitions

• A system is compatible if it has at least one solution.
• A system is incompatible if it has no solutions.
• A compatible system is determined if it has one solution and undetermined if it has multiple solutions.

Kronecker-Capelli Theorem

• A system is compatible if and only if the rank of the matrix of the system equals the rank of the augmented matrix.
  • $\text{rang}(A) = \text{rang}(\overline{A})$

Discussion of Compatibility and Number of Solutions

• Let $r = \text{rang}(A) = \text{rang}(\overline{A})$.
• If $r = n$, the system is compatible determined (one solution).
• If $r < n$, the system is compatible undetermined (infinitely many solutions).

Observations

• If $\text{rang}(A) < \text{rang}(\overline{A})$, the system is incompatible (no solutions).
• Homogeneous systems (all constants are zero) are always compatible.
• A system of $n$ equations with $n$ unknowns is a square system. If the determinant of the system's matrix is non-zero, the system is compatible and determined.

Statics

Force System Resultants

Main Topics

• Moment of a force.
• Principle of moments.
• Moment of a force about a specified axis.
• Simplification of a force and couple system.
• Further reduction of a force and couple system.

Moment of a Force Defined

• The moment of a force about a point measures the tendency of the force to cause rotation about that point.

Magnitude

• $M_O = Fd$, where:
  • $F$ = force magnitude
  • $d$ = moment arm (perpendicular distance)

Direction

• The moment vector $\overrightarrow{M_O}$ is perpendicular to the plane formed by the force $\overrightarrow{F}$ and moment arm $\overrightarrow{d}$.
• Direction is found using the right-hand rule.

Sense of Rotation

• Counterclockwise rotation: moment vector is directed out of the page.
• Clockwise rotation: moment vector is directed into the page.
• Right-hand rule: fingers curl in the direction of rotation, thumb points in the direction of the moment vector.

Varignon's Theorem

• The moment of a force is equal to the sum of the moments of the force's components: $\overrightarrow{M_O} = \overrightarrow{r} \times \overrightarrow{F} = \overrightarrow{r} \times (\overrightarrow{F_1} + \overrightarrow{F_2}) = \overrightarrow{r} \times \overrightarrow{F_1} + \overrightarrow{r} \times \overrightarrow{F_2}$

Algorithmic Game Theory

Definition

• Combines computer science (algorithm design and analysis) with game theory (mathematical study of strategic interactions).
• Uses computation to understand, predict, and design outcomes in strategic settings.

Selfish Routing

Model

• $G = (V, E)$: directed graph
• $r_e$: traffic rate on edge $e$
• $l_e(r_e)$: latency function for edge $e$
• $s_i, t_i$: source and sink for player $i$
• $\Gamma_i$: paths from $s_i$ to $t_i$
• $x_P$: flow of player $i$ along path $P \in \Gamma_i$
• $\sum_{P \in \Gamma_i} x_P = r_i$: total traffic of player $i$
• $r_e = \sum_{P: e \in P} x_P$: total traffic on edge $e$

Wardrop Equilibrium

• For all $i$ and all paths $P, P' \in \Gamma_i$ with $x_P > 0$: $\sum_{e \in P} l_e(r_e) \le \sum_{e \in P'} l_e(r_e)$. I.e., all flow is on min-latency paths.

Social Cost

• $SC(x) = \sum_{e \in E} r_e \cdot l_e(r_e)$ (total latency)

Price of Anarchy

• $PoA = \max_x \frac{SC(x)}{SC(x^)}$ where $x$ is Wardrop, $x^$ is socially optimal.

Braess's Paradox

• Adding an edge can worsen overall latency due to selfish behavior.

Theorem

• If latency functions are linear, $PoA \le \frac{4}{3}$.

The Chemistry of Life

The Elements of Life

• Life is mostly (96%) carbon, hydrogen, oxygen, and nitrogen.
• The remaining 4% mainly consists of calcium, phosphorus, potassium, and sulfur.
• Trace elements are required in very small amounts.

The Importance of Chemical Bonds

• An atom's characteristics determine how it interacts with others.
• Electronegativity is an atom’s attraction for electrons in a covalent bond.
  • Nonpolar covalent bond: electrons shared equally (similar electronegativity).
  • Polar covalent bond: unequal sharing (one atom is more electronegative).
• Ionic bond: attraction between ions.
• Van der Waals interactions: Weak attractions from transient partial charges.