Forces in 2D & Vector Notation

Unit Vector Notation: Forces are often written as $F = xi + yj$, where $i$ is a unit vector of magnitude 1 in the positive $x$-direction and $j$ is a unit vector of magnitude 1 in the positive $y$-direction.

Column Vector Notation: Alternatively, a force can be written as a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$, which clearly separates the horizontal and vertical components for easy calculation.

Resultant Force: The total force acting on a particle is the vector sum of all individual forces, calculated by adding the $i$ components together and the $j$ components together.

Calculating Magnitude: The magnitude of a force $F = xi + yj$ is its total length, found using Pythagoras' Theorem: $|F| = \sqrt{x^2 + y^2}$.

Calculating Direction: The direction is the angle $\theta$ the force makes with the positive $x$-axis, found using trigonometry: $\tan \theta = \frac{y}{x}$.

Quadrant Adjustments: When components are negative, the calculated angle $\tan^{-1}(\frac{|y|}{|x|})$ provides a reference angle; the final direction must be adjusted based on which quadrant the vector resides in (e.g., adding $180^\circ$ for a third-quadrant vector).

Condition for Equilibrium: A particle is in equilibrium if the resultant force acting on it is the zero vector, denoted as $0i + 0j$ or $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$.

Component Balance: For equilibrium to occur, the sum of all horizontal components must equal zero, and the sum of all vertical components must independently equal zero.

Geometric Interpretation: If the force vectors acting on a particle in equilibrium are drawn 'nose-to-tail', they will form a closed polygon, indicating no net displacement in force space.

Standardize Notation: While you can use either $i, j$ or column vectors, stick to one format throughout a single calculation to avoid transcription errors.

The Zero Vector: In equilibrium problems, always set the sum of components to zero. A common mistake is setting the sum of magnitudes to zero, which is incorrect.

Sanity Checks: Always verify the quadrant of your final direction. If $x$ is negative and $y$ is positive, your angle must be between $90^\circ$ and $180^\circ$.

Diagrams: Draw 'mini-diagrams' for individual forces to visualize their directions before combining them into a resultant vector.