Vector Physics Cheat Sheet

Cheatsheet Content

### Vector Basics - **Definition:** A quantity having both magnitude and direction. Represented as $\vec{A}$. - **Scalar:** A quantity having only magnitude (e.g., mass, time, temperature). - **Notation:** - Vector: $\vec{A}$ or $\mathbf{A}$ - Magnitude: $|\vec{A}|$ or $A$ - **Unit Vector:** A vector with magnitude 1, used to indicate direction. $\hat{u}_A = \frac{\vec{A}}{|\vec{A}|}$. - Cartesian unit vectors: $\hat{i}$ (x-direction), $\hat{j}$ (y-direction), $\hat{k}$ (z-direction). ### Vector Representation - **Component Form (2D):** $\vec{A} = A_x \hat{i} + A_y \hat{j}$ - **Component Form (3D):** $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ - **Magnitude:** - 2D: $|\vec{A}| = \sqrt{A_x^2 + A_y^2}$ - 3D: $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$ - **Direction (2D):** Angle $\theta$ with positive x-axis: $\tan\theta = \frac{A_y}{A_x}$. ### Vector Operations #### Addition & Subtraction - **Graphically:** Head-to-tail method for addition; $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$. - **Component-wise:** - If $\vec{A} = A_x \hat{i} + A_y \hat{j}$ and $\vec{B} = B_x \hat{i} + B_y \hat{j}$ - $\vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$ - $\vec{A} - \vec{B} = (A_x - B_x)\hat{i} + (A_y - B_y)\hat{j}$ #### Scalar Multiplication - $c\vec{A} = cA_x \hat{i} + cA_y \hat{j}$ - Multiplies magnitude by $|c|$ and reverses direction if $c ### Vector Calculus (Briefly) #### Position, Velocity, Acceleration - **Position Vector:** $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$ - **Velocity Vector:** $\vec{v}(t) = \frac{d\vec{r}}{dt} = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} + \frac{dz}{dt}\hat{k}$ - **Acceleration Vector:** $\vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}$ - **Speed:** $|\vec{v}(t)|$ #### Vector Fields - **Gradient:** $\nabla f = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} + \frac{\partial f}{\partial z}\hat{k}$ (Scalar to Vector) - **Divergence:** $\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$ (Vector to Scalar) - **Curl:** $\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix}$ (Vector to Vector)