When approaching forces in 2D using vector notation, follow these steps:

1. Express forces as vectors: Represent each force as a vector in component form, e.g., $\mathbf{F} = \begin{pmatrix} F_x \\ F_y \end{pmatrix}$ , where $F_x$ and $F_y$ are the horizontal and vertical components.
2. Resolve forces into components: For forces at an angle, resolve them into horizontal and vertical components using:

• $F_x = F \cos(\theta)$
• $F_y = F \sin(\theta)$

3. Apply Newton's second law: Use $\mathbf{F}_{\text{net}} = m\mathbf{a}$ to relate the net force to acceleration, with each force vector summed as:

• $\sum F_x = m a_x$
• $\sum F_y = m a_y$

4. Solve the vector equations: Find unknown forces or accelerations by solving the component equations. This method ensures a structured approach to handling forces using vector notation.

Example 1: Finding the Position Vector

Problem: A parallelogram $ABCD$ has vertices $A\left(\begin{array}{c}2 \\ 1 \\ 2\end{array}\right), B\left(\begin{array}{c}3 \\ 7 \\ 1\end{array}\right), C\left(\begin{array}{c}6 \\ 4 \\ -2\end{array}\right)$, and $D$ is unknown. Find the position vector of $D$.

Solution:

5. Use the property of parallelograms: In a parallelogram, opposite sides are equal and parallel. Thus, vector $\overrightarrow{BC} = \overrightarrow{AD}$.
6. Write the vector equation:

Simplifying:

7. Solve for $\overrightarrow{D}$:

Answer: The position vector of $D$ is $\overrightarrow{D} = \left(\begin{array}{c}5 \\ -8 \\ 7\end{array}\right)$.

Example 2: Magnitude of a Vector in 3D

Concept: The magnitude of a vector $\overrightarrow{a}$ in 3D is given by:

Problem: If $A\left(\begin{array}{c}2 \\7 \\ 9\end{array}\right) and B\left(\begin{array}{c}3 \\ -1 \\ 7\end{array}\right)$, find $|\overrightarrow{AB}|$.

Solution:

8. Find the vector $\overrightarrow{AB} = \overrightarrow{B} - \overrightarrow{A}$:

9. Calculate the magnitude $|\overrightarrow{AB}|$:

Answer: The magnitude $|\overrightarrow{AB}|$ is $\sqrt{69}$.

10. Resolve angled forces: Always break angled forces into horizontal and vertical components using $F_x = F \cos(\theta)$ and $F_y = F \sin(\theta)$.
11. Use vector form: Write forces as vectors $\mathbf{F} = \begin{pmatrix} F_x \\ F_y \end{pmatrix}$ to keep track of components easily.
12. Apply equilibrium or motion equations: Use $\sum F_x = m a_x$ and $\sum F_y = m a_y$ to set up equations for solving unknowns.