UPCAT Physics — Uncertainty in Measurement & VectorsCheat Sheet
Cheat sheet for UPCAT Physics — Uncertainty in Measurement & Vectors. Compact, printable, and organised around the concepts University of the Philippines tests most frequently in the UPCAT 2026. Perfect for the week before exam day.
Exam context
On the UPCAT 2026, the Physics subtest carries a "Core" weight in University of the Philippines's pattern. Uncertainty in Measurement & Vectors lands at position 1st out of 6 in the standard review order. Target score is UPG ≤ 2.2 typical, and roughly 20 items come from Physics on a typical UPCAT paper.
Uncertainty in Measurement & Vectors - Cheat sheet
Your last-minute revision companion for mastering measurement precision and vector operations before your UPCAT Physics exam.
Sections
Section Title
Significant Figures Rules
Important Facts
- Nonzero digits are ALWAYS significant
- Zeros between nonzeros are ALWAYS significant
- Leading zeros are NEVER significant
- Trailing zeros after decimal are ALWAYS significant
- Trailing zeros without decimal may or may not be significant
Key Definitions
Term
Significant Figures
Example
478 cm has 3 sig figs
Definition
Digits in a number that express it to the required level of accuracy
Term
Exact Numbers
Example
12 pieces in a dozen
Definition
Values with no uncertainty (definitions, counting, simple fractions)
Term
Inexact Numbers
Example
Height of a person
Definition
Values with uncertainty from measurements
Formulas
Formula
Multiplication/Division: Result = fewest sig figs
Meaning
Answer has same sig figs as the number with the least
Watch Out
Don't count decimal places, count significant digits
When To Use
When multiplying or dividing measured values
Formula
Addition/Subtraction: Result = fewest decimal places
Meaning
Answer rounded to position of least precise measurement
Watch Out
Look at decimal position, not total sig figs
When To Use
When adding or subtracting measured values
Section Title
Calculations with Significant Figures
Important Facts
- Scientific notation: N × 10^n where 1 ≤ |N| < 10
- All digits in scientific notation are significant
- Round final answer only, not intermediate steps
Key Definitions
Term
Accuracy
Example
Hitting the bullseye target
Definition
How close a measurement is to the actual value
Term
Precision
Example
Grouped arrows on target
Definition
How close repeated measurements are to each other
Formulas
Formula
Given unit × (desired unit/given unit) = desired unit
Meaning
Conversion factor method for unit changes
Watch Out
Make sure units cancel properly - desired unit in numerator
When To Use
Converting between different units
Section Title
Dimensional Analysis
Important Facts
- Also called Factor-Label Method or Unit Factor Method
- Conversion factors have value of 1
- Can chain multiple conversion factors together
Key Definitions
Term
Dimensional Analysis
Example
Converting meters to kilometers
Definition
Method to convert units using conversion factors
Term
Conversion Factor
Example
1000 m/1 km or 1 km/1000 m
Definition
Fraction where numerator and denominator represent same quantity in different units
Common Values
Value
meter
Symbol
m
Quantity
Length
Value
kilogram
Symbol
kg
Quantity
Mass
Value
second
Symbol
s
Quantity
Time
Section Title
Fundamental vs Derived Quantities
Important Facts
- 7 SI base units: length(m), mass(kg), time(s), current(A), temperature(K), amount(mol), luminous intensity(cd)
- All other units are combinations of base units
Key Definitions
Term
Fundamental Quantities
Example
Length, mass, time
Definition
Base units independent of other quantities
Term
Derived Quantities
Example
Density, area, volume
Definition
Quantities determined from fundamental quantities
Section Title
Scalar vs Vector Quantities
Important Facts
- Scalars add algebraically
- Vectors require special addition rules
- Speed is scalar, velocity is vector
- Distance is scalar, displacement is vector
Key Definitions
Term
Scalar Quantity
Example
Distance, mass, speed, temperature
Definition
Has magnitude only, no direction
Term
Vector Quantity
Example
Force, velocity, displacement, acceleration
Definition
Has both magnitude and direction
Term
Distance
Example
10 km traveled
Definition
Total ground covered during motion (scalar)
Term
Displacement
Example
5 km north of starting point
Definition
Change in position from start to end (vector)
Diagrams To Know
- Vector arrows showing magnitude and direction
- Vector addition diagrams (head-to-tail method)
Formulas
Formula
Same direction: R = A + B
Meaning
R is resultant, A and B are vector magnitudes
Watch Out
Direction of resultant is same as both vectors
When To Use
When vectors point in the same direction
Formula
Opposite direction: R = |A - B|
Meaning
R is magnitude difference, direction of larger vector
Watch Out
Take absolute value and direction of larger vector
When To Use
When vectors point in opposite directions
Formula
Perpendicular: R² = A² + B²
Meaning
Pythagorean theorem for perpendicular vectors
Watch Out
Only works for perpendicular vectors, not any angle
When To Use
When vectors are at 90° to each other
Formula
tan θ = opposite/adjacent
Meaning
θ is angle of resultant from reference axis
Watch Out
Make sure you identify correct opposite and adjacent sides
When To Use
Finding direction of resultant vector
Section Title
Vector Addition
Important Facts
- Vector addition is commutative: A + B = B + A
- Use head-to-tail method for graphical addition
- Components method works for any angle
Key Definitions
Term
Resultant Vector
Example
Net force from multiple forces
Definition
Single vector that represents the sum of two or more vectors
Diagrams To Know
- Vector addition triangles
- Component vector diagrams
- Right triangle for perpendicular vectors
Formulas
Formula
Direct: y = kx
Meaning
y and x increase/decrease together, k is constant
Watch Out
Graph is a straight line through origin
When To Use
When one quantity increases as another increases
Formula
Inverse: y = k/x
Meaning
y decreases as x increases, k is constant
Watch Out
Graph is a hyperbola, not a straight line
When To Use
When one quantity decreases as another increases
Section Title
Proportionality Relationships
Important Facts
- Direct proportion: ratio y/x is constant
- Inverse proportion: product xy is constant
Key Definitions
Term
Directly Proportional
Example
Distance and time at constant speed
Definition
When one quantity increases, the other increases at the same rate
Term
Inversely Proportional
Example
Speed and time for fixed distance
Definition
When one quantity decreases as the other increases at same rate
Diagrams To Know
- Direct proportion graph (straight line through origin)
- Inverse proportion graph (hyperbola)
Must Remember
- Leading zeros are NEVER significant, trailing zeros after decimal are ALWAYS significant
- Multiplication/division: count sig figs; Addition/subtraction: count decimal places
- Scientific notation N × 10^n where 1 ≤ |N| < 10
- Distance is scalar (total path), displacement is vector (change in position)
- Same direction vectors: add magnitudes; opposite: subtract magnitudes
- Perpendicular vectors: use Pythagorean theorem R² = A² + B²
- Direct proportion: y = kx (straight line); Inverse: y = k/x (hyperbola)
- Accuracy = closeness to true value; Precision = closeness to each other
- Conversion factor = (desired unit)/(given unit) so units cancel
- Vector addition is commutative: A + B = B + A
Last Minute Tips
- For sig figs: when in doubt, look at the measurement with the LEAST precision
- Vector problems: draw diagrams first - visualization prevents sign errors
- Dimensional analysis: write units for every number and make sure they cancel
- Don't confuse distance/displacement or speed/velocity - one has direction, one doesn't
- Perpendicular vector addition only works at 90° - use components method for other angles
Comparison Tables
Rows
Values
- Closeness to true value
- Closeness to each other
Property
Definition
Values
- Hitting bullseye
- Grouped arrows
Property
Target analogy
Values
- Correctness
- Repeatability
Property
Measurement focus
Columns
- Property
- Accuracy
- Precision
Table Title
Accuracy vs Precision
Rows
Values
- Scalar
- Vector
Property
Type
Values
- No direction
- Has direction
Property
Direction
Values
- Always positive
- Can be positive or negative
Property
Value
Values
- Depends on path taken
- Independent of path
Property
Path dependence
Columns
- Property
- Distance
- Displacement
Table Title
Distance vs Displacement
Rows
Values
- Magnitude only
- Magnitude and direction
Property
Components
Values
- Algebraic
- Geometric/Component method
Property
Addition
Values
- Mass, speed, temperature
- Force, velocity, displacement
Property
Examples
Columns
- Property
- Scalar
- Vector
Table Title
Scalar vs Vector
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