Vector Diagrams

Always use a sharp HB pencil, a ruler, and a protractor. Never rub out your construction lines, as they prove your method to the examiner.

Finding the Resultant Force

There are two main geometric methods to find the resultant of two forces.

The parallelogram of forces method is used when two forces act from the exact same starting point (tail-to-tail). You draw dashed lines parallel to each vector to form a four-sided shape, then draw the resultant force as the solid diagonal line originating from the starting point.

The tip-to-tail method (or triangle method) is used when you place the tail of the second vector at the tip of the first. The resultant is the straight line drawn directly from the start of the very first vector to the end of the final vector.

Worked Example: Parallelogram Method

Two forces, $60\text{ N}$ and $80\text{ N}$, act on an object at an angle of $45^\circ$ to each other. Describe how to find the resultant force.

Step 1: Choose a sensible scale.

• Let $1\text{ cm} = 10\text{ N}$.

Step 2: Draw the original forces tail-to-tail.

• Draw an $8\text{ cm}$ horizontal line.
• From the exact same starting origin, draw a $6\text{ cm}$ line at an angle of $45^\circ$ using a protractor.

Step 3: Construct the parallelogram.

• From the tip of the $6\text{ cm}$ line, draw a dashed construction line parallel to the $8\text{ cm}$ line.
• From the tip of the $8\text{ cm}$ line, draw a dashed construction line parallel to the $6\text{ cm}$ line.

Step 4: Draw and measure the resultant.

• Draw a solid line from the starting origin to the point where the dashed lines intersect.
• Measure its length ($13.0\text{ cm}$, which equals $130\text{ N}$) and its angle (approximately $19^\circ$ from the $80\text{ N}$ force).

Resolving a Single Force

Instead of combining forces together, you can do the exact reverse by resolving a force. This is the process of splitting a single diagonal force into two perpendicular parts, known as components (usually horizontal and vertical).

To resolve a force graphically, draw the original force to scale at the correct angle. Then, form a right-angled triangle around it by dropping vertical and horizontal lines from the tip down to the axes. Measure the lengths of these horizontal and vertical sides and use your scale to convert them back into newtons.

Alternatively, you can calculate the components mathematically using trigonometry. If a force $F$ acts at an angle $\theta$ to the horizontal, the components are:

Equilibrium in Vector Diagrams

An object is in equilibrium when the resultant force acting on it is exactly zero. According to Newton's First Law, an object in a state of equilibrium will either be perfectly stationary or moving at a constant velocity in a straight line.

In a vector diagram, equilibrium is represented by a closed loop. If you draw all the forces acting on an object tip-to-tail, the tip of the final arrow will perfectly meet the exact starting tail of the first arrow, forming a completely closed shape (like a triangle for three forces).