Finals - Internet Research for Business

Matrices

Basic Definitions

• A matrix $A$ is a table of numbers with $m$ horizontal rows and $n$ vertical columns.
• $A$ is represented as: $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$
• An alternate representation is $A = (a_{ij})_{m \times n}$.
• The size of the matrix is $m \times n$.
• An element is a single number $a_{ij}$.
• Index $i$ indicates the horizontal row and $j$ indicates the vertical column.
• A square matrix has an equal number of rows and columns $m = n$.

Example

• An example of a $2 \times 3$ matrix is: $A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$

Matrix Addition and Scalar Multiplication

• Matrices $A$ and $B$ are the same size if they have the same number of horizontal and vertical rows
• Matrices of the same size can be added together.
• For $A = (a_{ij}){m \times n}$ and $B = (b{ij}){m \times n}$, $A + B = (a{ij} + b_{ij})_{m \times n}$.
• A matrix can be multiplied by a real number (scalar).
• If $A = (a_{ij}){m \times n}$ and $c \in \mathbb{R}$, then $cA = (ca{ij})_{m \times n}$.

Example

• $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1+5 & 2+6 \ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$
• $2\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 2 \cdot 1 & 2 \cdot 2 \ 2 \cdot 3 & 2 \cdot 4 \end{bmatrix} = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}$

Physics

Vectors

Vector Sum

Analytical Method

• Vector components are $\overrightarrow{A} = A_x\hat{i} + A_y\hat{j}$
• $A_x = A \cos \theta$
• $A_y = A \sin \theta$
• The resultant vector is $\overrightarrow{R} = \overrightarrow{A} + \overrightarrow{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$
• $R_x = A_x + B_x$
• $R_y = A_y + B_y$
• Magnitude of resultant vector is $R = \sqrt{R_x^2 + R_y^2}$
• Angle is $\theta = \tan^{-1} (\frac{R_y}{R_x})$

Scalar Product (Dot Product)

• $\overrightarrow{A} \cdot \overrightarrow{B} = AB \cos \theta$
• $\overrightarrow{A} \cdot \overrightarrow{B} = (A_x\hat{i} + A_y\hat{j}) \cdot (B_x\hat{i} + B_y\hat{j}) = A_xB_x + A_yB_y$

Vector Product (Cross Product)

• $\overrightarrow{A} \times \overrightarrow{B} = AB \sin \theta \hat{n}$
• $\overrightarrow{A} \times \overrightarrow{B} = (A_x\hat{i} + A_y\hat{j} + A_z\hat{k}) \times (B_x\hat{i} + B_y\hat{j} + B_z\hat{k}) = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$
• $\overrightarrow{A} \times \overrightarrow{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix}$

Kinematics

Uniform Rectilinear Motion (MRU)

• $v = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$
• $x_f = x_i + v(t_f - t_i)$

Uniformly Varied Rectilinear Motion (MRUV)

• $a = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}$
• $v_f = v_i + a(t_f - t_i)$
• $x_f = x_i + v_i(t_f - t_i) + \frac{1}{2}a(t_f - t_i)^2$
• $v_f^2 = v_i^2 + 2a(x_f - x_i)$

Free Fall and Vertical Throw

• $a = -g = -9.8 m/s^2$
• $v_f = v_i - g(t_f - t_i)$
• $y_f = y_i + v_i(t_f - t_i) - \frac{1}{2}g(t_f - t_i)^2$
• $v_f^2 = v_i^2 - 2g(y_f - y_i)$

Oblique Throw

• $v_{0x} = v_0 \cos \theta$
• $v_{0y} = v_0 \sin \theta$
• $x_f = x_i + v_{0x}(t_f - t_i)$
• $y_f = y_i + v_{0y}(t_f - t_i) - \frac{1}{2}g(t_f - t_i)^2$
• $v_{fx} = v_{0x}$
• $v_{fy} = v_{0y} - g(t_f - t_i)$
• Time to reach max height $t_{v} = \frac{v_{0y}}{g}$
• Total time of flight $t_{total} = 2t_v = \frac{2v_{0y}}{g}$
• Maximum horizontal distance $x_{max} = \frac{v_0^2 \sin 2\theta}{g}$
• Maximum height reached $y_{max} = \frac{v_0^2 \sin^2 \theta}{2g}$

Uniform Circular Motion (MCU)

• Angular velocity $\omega = \frac{\Delta \theta}{\Delta t}$
• Tangential velocity $v = \omega r$
• Centripetal acceleration $a_c = \frac{v^2}{r} = \omega^2 r$
• Period $T = \frac{2\pi r}{v} = \frac{2\pi}{\omega}$
• Frequency $f = \frac{1}{T} = \frac{\omega}{2\pi}$

Uniformly Varied Circular Motion (MCUV)

• Angular acceleration $\alpha = \frac{\Delta \omega}{\Delta t}$
• $\omega_f = \omega_i + \alpha(t_f - t_i)$
• $\theta_f = \theta_i + \omega_i(t_f - t_i) + \frac{1}{2}\alpha(t_f - t_i)^2$
• $\omega_f^2 = \omega_i^2 + 2\alpha(\theta_f - \theta_i)$