Basic Vectors

• A vector is a quantity defined by its magnitude (length or size) and its direction in space. In 2D, this is often visualized as an arrow connecting a starting point to an ending point.

• Vectors are typically denoted using bold lowercase letters like $\mathbf{a}$ or $\mathbf{b}$ in print, or by underlining them ($\underline{a}$) when handwritten. If the start and end points are known, such as $A$ and $B$, the vector is written as $\vec{AB}$.

• The zero vector, denoted as $\mathbf{0}$, has a magnitude of zero and no specific direction, representing a lack of movement or a state of equilibrium.

• Column Vectors: Represented as $\begin{pmatrix} x \\ y \end{pmatrix}$, where $x$ indicates horizontal displacement (right is positive) and $y$ indicates vertical displacement (up is positive).

• Unit Vector Notation: Uses the standard basis vectors $\mathbf{i}$ and $\mathbf{j}$, where $\mathbf{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\mathbf{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. A vector is written as $x\mathbf{i} + y\mathbf{j}$.

• Magnitude Calculation: The magnitude of a vector $\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}$ is found using Pythagoras' Theorem: $|\mathbf{a}| = \sqrt{x^2 + y^2}$. This value is always a non-negative scalar.

• Scalar Multiplication: Multiplying a vector by a positive scalar $k$ changes its magnitude by a factor of $k$ but preserves its direction. If $k$ is negative, the direction is reversed.

• Parallelism: Two vectors are parallel if and only if one is a scalar multiple of the other ($\mathbf{a} = k\mathbf{b}$). This means they share the same line of action or orientation, regardless of their magnitude.

• Unit Vectors: A unit vector is any vector with a magnitude of exactly 1. To find a unit vector in the direction of $\mathbf{a}$, divide the vector by its own magnitude: $\mathbf{\hat{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$.

• Resultant Vectors: The sum of two or more vectors is called the resultant. Geometrically, this is found by placing vectors 'nose-to-tail'; the resultant is the vector from the start of the first to the end of the last.

• Algebraic Addition: To add vectors in component form, simply add the corresponding $x$ and $y$ values: $\begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}$.

• Subtraction: Subtracting a vector $\mathbf{b}$ from $\mathbf{a}$ is equivalent to adding the negative of $\mathbf{b}$ ($\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b})$), which geometrically represents the vector pointing from the tip of $\mathbf{b}$ to the tip of $\mathbf{a}$ when both start at the same point.

Key Distinctions

• To find the displacement vector between two points $A$ and $B$, subtract the position vector of the start from the position vector of the end: $\vec{AB} = \vec{OB} - \vec{OA}$.