In classical physics, physical quantities are categorized based on their directional properties. Distinguishing between scalar and vector quantities is essential for analyzing multidimensional forces, orbital motion, and field theory in Cambridge IGCSE Physics.

A scalar quantity is defined completely by its magnitude (numerical size) and appropriate physical unit. Scalars obey ordinary arithmetic rules of addition and subtraction. Examples of scalars include:

• Distance ($s$)
• Speed ($v$)
• Time ($t$)
• Mass ($m$)
• Energy ($E$)
• Temperature ($\theta$ or $T$)

A vector quantity is defined by both a magnitude and a direction in space. Vectors cannot be combined by simple scalar addition unless they act along the exact same line. Examples of vectors include:

• Displacement ($s$)
• Velocity ($v$)
• Acceleration ($a$)
• Force ($F$) and Weight ($W$)
• Momentum ($p$)
• Electric Field Strength ($E$)
• Gravitational Field Strength ($g$)

1. Vector Representation

Vectors are represented visually by straight arrows:

• The length of the arrow is proportional to the vector's magnitude.
• The orientation of the arrowhead specifies the vector's direction.

2. Resultant of Two Perpendicular Vectors

When two vector quantities (such as two perpendicular forces $F_x$ and $F_y$, or perpendicular velocity components $v_x$ and $v_y$) act simultaneously on a single body, their combined effect is represented by a single resultant vector ($R$).

To determine the magnitude of the resultant vector at a $90^\circ$ angle, apply Pythagoras' Theorem:

To determine the direction angle ($\theta$) relative to the horizontal component:

3. Graphical Vector Addition (Tip-to-Tail Method)

1. Draw the first vector $\vec{A}$ to scale in its specified direction.
2. Draw the second vector $\vec{B}$ starting from the tip (arrowhead) of $\vec{A}$.
3. Draw the resultant vector $\vec{R}$ from the tail (start) of $\vec{A}$ directly to the tip of $\vec{B}$.
4. Measure the length of $\vec{R}$ with a ruler to find magnitude, and measure angle $\theta$ with a protractor.

Vector mechanics is critical in aviation navigation and structural engineering. Aircraft flight computers continuously calculate wind vector triangles: the plane's airspeed vector added to the wind velocity vector yields the ground track vector. In civil engineering, truss bridges are designed by calculating vector components to ensure structural equilibrium under dynamic loads.

Understanding vectors elevates physics from one-dimensional arithmetic to multi-dimensional spatial analysis. Utilizing Pythagoras' theorem and trigonometric ratios allows precise calculation of resultant velocities and forces in complex mechanical systems.