IMAT Physics · Measurement and Units

Dimensional Analysis, Scalars and Vectors

Both sides of a true equation, and every term in a sum, carry the same units; some quantities also carry a direction, and those add by geometry, not arithmetic.

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Why this matters

Four of the five past questions in this chapter are on this page. The 2013, 2015 and 2021 papers asked which expressions or equations have consistent units, and the 2012 paper asked which quantity in a list is not a vector.

Concept 1 of 3: Dimensional analysis: checking units in an equation

You cannot add 3 metres to 2 seconds. In the same way, a correct physics equation can only add, subtract or equate things measured in the same units. Breaking every unit down to kilograms, metres, seconds and amperes lets you test an equation, or name the quantity an expression gives, without knowing any numbers.

Definition

Dimensional analysis checks units by writing every quantity in SI base units.

Quantities can be added, subtracted or set equal only if they have the same units.
So every term of a sum must have the same units as the left-hand side.
Pure numbers ( $\tfrac{1}{2}$ , $2\pi$ ) have no units, so a units check can never show a missing factor of $\tfrac{1}{2}$ .
Useful chains: energy = force × distance = power × time; power = force × velocity; charge = current × time; $V = IR$ and $P = VI = V^2/R$ .

Working a unit down to base units

\text{J} = \text{N m} = (\text{kg m s}^{-2})(\text{m}) = \text{kg m}^2\,\text{s}^{-2}

$\text{J}$ joule, the unit of energy and work
$\text{N}$ newton, the unit of force

Worked example

P

is a power,

t

a time and

F

a force. Show that

\dfrac{Pt}{F}

has the units of a length. Then check whether

s = ut + \tfrac{1}{2}at^2

has consistent units.

Practice this conceptself-check · 4 quick reps

Consistent units do not prove an equation is right

s = ut + at^2

has consistent units but is wrong: the factor

\tfrac{1}{2}

is missing. A units check can rule an equation out, but it can never confirm a numerical factor.

A resistance and its reciprocal cannot be added

1/R

is measured in

\Omega^{-1}

, so an expression like

R_1 + 1/R_2

mixes units and cannot be right. The same goes for any sum that mixes, say, an energy and a power.

Concept 2 of 3: Scalars and vectors: which quantities have a direction

Some quantities are fully described by a size: 5 kg of flour is 5 kg in any direction. Others make no sense without a direction: a velocity of 20 m/s could be north or south, and that changes everything. The test is simple: ask whether turning the quantity around would change its meaning.

Definition

A scalar has a size (magnitude) only. Scalars add like ordinary numbers.
A vector has a size and a direction. Vectors add by geometry (next concept).
A negative sign on a scalar (a charge of $-2\ \mu\text{C}$ , a temperature of $-5$ °C) is not a direction.
Every force is a vector, so weight (a force) is a vector, but mass is a scalar.

Quantity	Scalar or vector	Note
Distance	scalar	length of the path travelled
Displacement	vector	straight-line change of position
Speed	scalar	distance per unit time
Velocity	vector	displacement per unit time
Acceleration	vector	change of velocity per unit time
Mass	scalar	amount of matter
Weight, any force	vector	weight points towards the centre of the Earth
Momentum	vector	same direction as the velocity
Energy, work, power	scalar	no direction, even though forces cause them
Pressure	scalar	the force it makes on a surface is a vector
Temperature	scalar	a minus sign is not a direction
Electric charge	scalar	positive or negative, never a direction
Electric field	vector	direction of the force on a positive charge
Electric current	scalar	currents at a junction add as numbers

Practice this conceptself-check · 3 quick reps

Mass is a scalar, weight is a vector

Weight is the gravitational force on a mass, so it has a direction (down). Mass has none. An option listing weight as a scalar confuses the two.

Concept 3 of 3: Adding vectors and splitting them into components

Walk 5 m east and then 12 m north: you are not 17 m from the start, because the two steps point in different directions. Vectors add tip to tail, and the result is the arrow from the first tail to the last tip. When two vectors are at right angles, Pythagoras gives its length. Any vector can also be split into two perpendicular parts, which is how most problems are solved.

Definition

Resultant: the single vector with the same effect as several vectors together (draw them tip to tail).
Two perpendicular vectors $a$ and $b$ : resultant $\sqrt{a^2 + b^2}$ , at angle $\tan^{-1}(b/a)$ to $a$ .
Two vectors at any angle: the resultant lies between $|a - b|$ (opposite) and $a + b$ (same direction).
Components of a vector $F$ at angle $\theta$ to the horizontal: $F\cos\theta$ horizontally and $F\sin\theta$ vertically.
To add many vectors, add all horizontal components, add all vertical components, then combine the two totals with Pythagoras.

Components and the resultant

F_x = F\cos\theta, \quad F_y = F\sin\theta, \quad R = \sqrt{F_x^2 + F_y^2}

$\theta$ angle between the vector and the horizontal
$F_x, F_y$ horizontal and vertical components
$R$ size of the resultant

Worked example

Two forces act on a point: 5.0 N towards the east and 12 N towards the north. Find the size and direction of the resultant.

Practice this conceptself-check · 3 quick reps

Vector sizes add only when the vectors are parallel

3 N and 4 N give 7 N only if they point the same way. At right angles they give 5 N, and in opposite directions 1 N. The resultant is always between the difference and the sum of the sizes.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Dimensional analysis: checking units in an equation
Working a unit down to base units
$\text{J} = \text{N m} = (\text{kg m s}^{-2})(\text{m}) = \text{kg m}^2\,\text{s}^{-2}$
Adding vectors and splitting them into components
Components and the resultant
$F_x = F\cos\theta, \quad F_y = F\sin\theta, \quad R = \sqrt{F_x^2 + F_y^2}$

Reference tables (1)

Scalars and vectors: which quantities have a direction14 rows

Quantity	Scalar or vector	Note
Distance	scalar	length of the path travelled
Displacement	vector	straight-line change of position
Speed	scalar	distance per unit time
Velocity	vector	displacement per unit time
Acceleration	vector	change of velocity per unit time
Mass	scalar	amount of matter
Weight, any force	vector	weight points towards the centre of the Earth
Momentum	vector	same direction as the velocity
Energy, work, power	scalar	no direction, even though forces cause them
Pressure	scalar	the force it makes on a surface is a vector
Temperature	scalar	a minus sign is not a direction
Electric charge	scalar	positive or negative, never a direction
Electric field	vector	direction of the force on a positive charge
Electric current	scalar	currents at a junction add as numbers

Watch out for (4)

Consistent units do not prove an equation is right
→ Dimensional analysis: checking units in an equation
A resistance and its reciprocal cannot be added
→ Dimensional analysis: checking units in an equation
Mass is a scalar, weight is a vector
→ Scalars and vectors: which quantities have a direction
Vector sizes add only when the vectors are parallel
→ Adding vectors and splitting them into components

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