Cambridge A-Level Physics 9702 – Scalars and \cdot ectors
Scalars and \cdot ectors
Objective
By the end of this lesson students will be able to:
Define scalar and vector quantities.
Distinguish between scalars and vectors using their mathematical properties.
Identify common scalar and vector quantities that appear in the Cambridge A‑Level Physics (9702) syllabus.
Represent vectors graphically and algebraically.
Definitions
A scalar quantity is described completely by a magnitude (numerical value) and a unit. It has no direction.
Examples include mass, temperature, energy, and speed.
A vector quantity has both magnitude and direction. It is represented by an arrow, a bold typeface, or a symbol with an arrow over it, e.g. \$\mathbf{v}\$ or \$\vec{v}\$.
Mathematical Representation
For a vector \$\vec{A}\$ with components in a Cartesian coordinate system:
\$\vec{A}=Ax\hat{i}+Ay\hat{j}+A_z\hat{k}\$
The magnitude (or length) of \$\vec{A}\$ is given by:
\$|\vec{A}|=\sqrt{Ax^{2}+Ay^{2}+A_z^{2}}\$
Scalars are denoted simply by a symbol, e.g. \$m\$, \$T\$, \$E\$, without any directional notation.
Graphical Representation
Suggested diagram: Arrow representing a vector with its magnitude labelled and direction indicated by the arrowhead.
Comparison of Scalars and \cdot ectors
Property
Scalar
Vector
Definition
Magnitude only
Magnitude and direction
Notation
Italic letters, e.g. \$m\$, \$T\$
Bold or arrow, e.g. \$\mathbf{v}\$, \$\vec{v}\$
Addition
Simple arithmetic: \$a+b\$
Vector addition (head‑to‑tail or componentwise)
Multiplication
Scalar × scalar = scalar
Scalar × vector = vector; vector × vector can give scalar (dot product) or vector (cross product)
Units
Units of the quantity alone (e.g., kg, J, K)
Units of the quantity plus direction (e.g., m s⁻¹, N, T)
Examples from the Cambridge A‑Level Physics (9702) Syllabus
Scalar Quantities
Mass (\$m\$) – kg
Temperature (\$T\$) – K or °C
Energy (\$E\$, \$K\$, \$U\$) – J
Speed (\$s\$) – m s⁻¹
Work done (\$W\$) – J
Power (\$P\$) – W
Electric charge (\$Q\$) – C
Electric potential difference (\$V\$) – V
Pressure (\$p\$) – Pa
Vector Quantities
Displacement (\$\vec{s}\$) – m
Velocity (\$\vec{v}\$) – m s⁻¹
Acceleration (\$\vec{a}\$) – m s⁻²
Force (\$\vec{F}\$) – N
Momentum (\$\vec{p}\$) – kg m s⁻¹
Weight (\$\vec{W}\$) – N (direction toward Earth’s centre)
Electric field (\$\vec{E}\$) – N C⁻¹
Magnetic field (\$\vec{B}\$) – T
Torque (\$\vec{\tau}\$) – N m
Key Points to Remember
All vector quantities can be resolved into perpendicular components; scalars cannot.
When adding or subtracting vectors, both magnitude and direction must be considered.
Physical laws often involve vectors (e.g., Newton’s second law \$\vec{F}=m\vec{a}\$) whereas derived quantities such as kinetic energy \$K=\frac{1}{2}mv^{2}\$ are scalars.
In examinations, be careful to indicate direction for vectors and to keep units consistent.