understand the difference between scalar and vector quantities and give examples of scalar and vector quantities included in the syllabus
Scalars and Vectors – Cambridge International AS & A Level Physics (9702)
Learning Objectives (AO1‑AO3)
- AO1 – Knowledge: Define scalar and vector quantities; state their distinguishing properties and give syllabus‑based examples.
- AO2 – Application: Represent vectors graphically and algebraically (including unit‑vector notation); resolve vectors into perpendicular components; add vectors using both graphical and component methods; apply dot‑ and cross‑product formulas.
- AO3 – Analysis: Use vector notation correctly in physical laws; propagate uncertainties for both magnitude and direction of vector quantities; recognise when a result is a scalar (dot product) or a vector (cross product).
Syllabus Coverage
This note satisfies the following sub‑sections of the 9702 syllabus:
- 1.4 Scalars and vectors – definitions, notation, graphical representation, addition, multiplication.
- 1.5 Error analysis – estimating uncertainties for vector quantities.
- 2.1 Kinematics – displacement, velocity, acceleration (vectors).
- 2.2 Forces – Newton’s laws, momentum, impulse (vectors).
- 2.3 Turning effects of forces – torque (vector).
- 2.4 Electric and magnetic fields – vector nature of E and B.
Definitions
| Scalar quantity | Described completely by a magnitude (numerical value) and a unit. No direction is associated. |
|---|---|
| Vector quantity | Has both magnitude and a specific direction. Written in bold, with an arrow, or using unit‑vector notation (e.g. v, \(\vec v\), \(\mathbf{v}\)). |
Notation
| Type | Symbol | Typical Units |
|---|---|---|
| Scalar | italic letters, e.g. m, T, E | kg, K, J, m s⁻¹ … |
| Vector | bold or arrow, e.g. v, \(\vec v\), \(\mathbf{v}\) | m s⁻¹, N, T … (same units as the magnitude, direction indicated separately) |
Unit‑vector notation
In a Cartesian system the unit vectors \(\hat{\imath},\hat{\jmath},\hat{k}\) point along the +x, +y and +z axes respectively. Any vector \(\vec A\) can be written as
\[
\vec A = Ax\hat{\imath}+Ay\hat{\jmath}+A_z\hat{k}
\]
where \(Ax, Ay, A_z\) are the scalar components of \(\vec A\) along the three axes.
Mathematical Representation of a Vector
Magnitude (length) of \(\vec A\):
\[
|\vec A|=\sqrt{Ax^{2}+Ay^{2}+A_z^{2}}
\]
Direction is often expressed as an angle \(\theta\) measured from a chosen axis, or as a unit‑vector direction cosines.
Graphical Representation
Resolving a Vector into Perpendicular Components
Any planar vector can be expressed as the sum of two perpendicular components, usually along the x- and y-axes.
Worked example (2‑D)
Resolve a displacement of 12 m at \(30^{\circ}\) north of east.
\[
\begin{aligned}
\vec s &= sx\hat{\imath}+sy\hat{\jmath}\\
s_x &= s\cos\theta = 12\cos30^{\circ}=10.4\ \text{m (east)}\\
s_y &= s\sin\theta = 12\sin30^{\circ}=6.0\ \text{m (north)}
\end{aligned}
\]
\[
\vec s = 10.4\hat{\imath}+6.0\hat{\jmath}\ \text{m}
\]
Worked example (3‑D)
A particle moves 5 m at an azimuth of \(45^{\circ}\) (measured clockwise from +x) and an elevation of \(30^{\circ}\) above the horizontal.
\[
\begin{aligned}
s_x &= s\cos\!30^{\circ}\cos\!45^{\circ}=5(0.866)(0.707)=3.06\ \text{m}\\
s_y &= s\cos\!30^{\circ}\sin\!45^{\circ}=5(0.866)(0.707)=3.06\ \text{m}\\
s_z &= s\sin\!30^{\circ}=5(0.500)=2.50\ \text{m}
\end{aligned}
\]
\[
\vec s = 3.06\hat{\imath}+3.06\hat{\jmath}+2.50\hat{k}\ \text{m}
\]
Vector Addition
- Graphical (head‑to‑tail) method – Place the tail of the second vector at the head of the first; the resultant runs from the tail of the first to the head of the second.
- Component (algebraic) method – Add the corresponding components:
\[
\vec R = (Ax+Bx)\hat{\imath}+(Ay+By)\hat{\jmath}+(Az+Bz)\hat{k}
\]
Numeric example
\[
\vec A = 3\hat{\imath}+4\hat{\jmath}\ \text{m},\qquad
\vec B = -2\hat{\imath}+5\hat{\jmath}\ \text{m}
\]
\[
\vec R = (3-2)\hat{\imath}+(4+5)\hat{\jmath}=1\hat{\imath}+9\hat{\jmath}\ \text{m}
\]
\[
|\vec R|=\sqrt{1^{2}+9^{2}}=9.06\ \text{m},\qquad
\theta = \tan^{-1}\!\left(\frac{9}{1}\right)=84^{\circ}\ \text{above the +x‑axis}
\]
Vector Multiplication
- Dot product (scalar product):
\[
\vec A\!\cdot\!\vec B = |\vec A|\,|\vec B|\cos\theta
\]
Result is a scalar. Physics example: work done, \(W = \vec F\!\cdot\!\vec s\).
- Cross product (vector product):
\[
\vec A\!\times\!\vec B = |\vec A|\,|\vec B|\sin\theta\,\hat n
\]
Result is a vector perpendicular to the plane of \(\vec A\) and \(\vec B\). Physics examples: torque \(\vec\tau = \vec r\!\times\!\vec F\) and magnetic force \(\vec F = q\vec v\!\times\!\vec B\).
Worked problem – Torque (cross product)
A force of 5 N acts at the end of a 0.4 m lever that is inclined \(30^{\circ}\) above the horizontal. The force is applied perpendicular to the lever (i.e. at \(90^{\circ}\) to the lever). Find the magnitude and direction of the torque about the pivot.
\[
\begin{aligned}
|\vec\tau| &= rF\sin\phi \\
&= (0.40\ \text{m})(5\ \text{N})\sin 90^{\circ} \\
&= 2.0\ \text{N m}
\end{aligned}
\]
Direction: using the right‑hand rule, the torque points out of the page (positive \(\hat{k}\)) for a counter‑clockwise rotation.
Physics Laws that Involve Vectors
- Newton’s second law: \(\displaystyle \vec F = m\vec a\)
- Momentum: \(\displaystyle \vec p = m\vec v\)
- Impulse–momentum theorem: \(\displaystyle \Delta\vec p = \vec F\Delta t\)
- Work–energy theorem (scalar result): \(\displaystyle W = \vec F\!\cdot\!\vec s\)
- Torque: \(\displaystyle \vec\tau = \vec r\!\times\!\vec F\)
- Electric field: \(\displaystyle \vec E = \frac{\vec F}{q}\)
- Magnetic force on a moving charge: \(\displaystyle \vec F = q\vec v\!\times\!\vec B\)
Examples from the Cambridge AS & A Level Physics (9702) Syllabus
| Scalar Quantities (AO1) | Typical Units | Vector Quantities (AO1) | Typical Units |
|---|---|---|---|
| Mass (\(m\)) | kg | Displacement (\(\vec s\)) | m |
| Temperature (\(T\)) | K or °C | Velocity (\(\vec v\)) | m s⁻¹ |
| Energy (\(E, K, U\)) | J | Acceleration (\(\vec a\)) | m s⁻² |
| Speed (\(s\)) | m s⁻¹ | Force (\(\vec F\)) | N |
| Work (\(W\)) | J | Momentum (\(\vec p\)) | kg m s⁻¹ |
| Power (\(P\)) | W | Weight (\(\vec W\)) | N (toward Earth’s centre) |
| Electric charge (\(Q\)) | C | Electric field (\(\vec E\)) | N C⁻¹ |
| Potential difference (\(V\)) | V | Magnetic field (\(\vec B\)) | T |
| Pressure (\(p\)) | Pa | Torque (\(\vec\tau\)) | N m (direction given by right‑hand rule) |
Error Analysis for Vector Quantities (AO3)
- Magnitude uncertainty – Propagate using standard rules. For a 2‑D vector \(|\vec A|=\sqrt{Ax^{2}+Ay^{2}}\):
\[
\frac{\Delta|\vec A|}{|\vec A|}= \frac{1}{|\vec A|}\sqrt{(Ax\Delta Ax)^{2}+(Ay\Delta Ay)^{2}}
\]
- Direction uncertainty – If the angle \(\theta\) is measured, quote \(\Delta\theta\) (e.g. \(\pm1^{\circ}\) for a typical protractor).
- Combine both to give a full statement, e.g.
\[
\vec F = (3.2\pm0.2)\,\text{N at }45^{\circ}\pm2^{\circ}
\]
Key Points to Remember (ordered by AO level)
- AO1 – Knowledge
- Scalars: magnitude only.
- Vectors: magnitude + direction.
- Notation: italics for scalars; bold, arrow, or unit‑vector form for vectors.
- Typical syllabus examples are listed in the table above.
- AO2 – Application
- Resolve vectors into perpendicular components using trigonometry (2‑D and 3‑D).
- Add vectors graphically (head‑to‑tail) and algebraically (component‑wise).
- Apply dot product for work, energy, power; apply cross product for torque and magnetic force.
- AO3 – Analysis
- Insert vectors correctly into physical laws (e.g. \(\vec F=m\vec a\), \(\vec\tau=\vec r\times\vec F\)).
- Propagate uncertainties for both magnitude and direction of vector results.
- Remember that units of a vector are the same as those of its magnitude; direction is expressed separately (e.g. “N m counter‑clockwise”).