Home

Physics

Motion

Resolving vectors

Select Lesson

Exam Board

Select an option

Oscillations

Simple harmonic motion

Displacement, velocity and acceleration for simple harmonic motion

Energy within simple harmonic motion

Damping and resonance

Simple harmonic oscillator and pendulum

Gravitational Fields

Gravitational fields

The law of gravitation

Kepler's laws

Gravitational potential and gravitational potential energy

Nuclear physics

Einstein's mass-energy equation

Nuclear fission and nuclear reactors

Nuclear fusion

Radioactivity

Radioactivity: alpha, beta and gamma radiation

Half-life and radioactive decay

Cosmology

Units for astronomical distances

Hubble's law

The evolution of the Universe

Thermal physics

Measuring temperature and temperature scales

Solids, liquids and gases

Internal energy

Specific latent heat and specific heat capacity

Ideal gases

The ideal gas law

Kinetic theory of gases: Root mean square speed

Black-body radiaton and stellar luminosity

Particle physics

The nuclear model of the atom

Magnitudes of the atom

Fundamental particles

Quarks and anti-quarks

Particle accelerators and detectors

Electromagnetism

Magnetic fields and electromagnetism

Magnetic flux density

Charged particles in a magnetic field

Faraday's and Lenz's Laws

Uses of electromagnetic induction

Alternating current and voltage

Capacitance

Capacitors and capacitance

Energy stored in a capacitor

Charging and discharging capacitors

Electric fields

Coulomb's law

Electric field between two parallel plates

Motion of charged particles in an electric field

Electric potential and potential energy

Circular Motion

Angular velocity and the radian

Centripetal acceleration

Centripetal force

Quantum physics

The photon model

The photoelectric effect

The photoelectric effect equation

Wave-particle duality

Energy levels and spectra

Lenses

Power of a lens

Ray diagrams, the thin lens equation and magnification

Waves

Progressive waves

Waves: displacement-time and distance-time graphs

Refraction and Snell's law

Reflection and the critical angle

Intensity of a wave

Diffraction and polarisation

The principle of superposition

Young's double-slit experiment

Stationary waves

Harmonics

Materials

Hooke's law

Elastic and plastic deformation

Stress, strain and Young's modulus

Stress-strain graphs

Fluids

Density and pressure

Fluid flow and viscosity

Terminal velocity

Electrical circuits

Circuits: diagrams and components

Potential difference and electromotive force

Resistance and Ohm's Law

Resistivity

I-V characteristics

Conductors and semiconductors

Combining resistors and Kirchhoff's Laws

Internal resistance

Potential dividers

Current and charge

Current and charges

Conventional current and electron flow

Mean drift velocity

Newton's laws of motion and momentum

Momentum: definition, conservation and calculations

Collisions in two dimensions

Newton's laws of motion

Energy

Forms of energy, conservation and work done

Kinetic energy and gravitational potential energy

Power and efficiency

Motion

Scalar and vector quantities

Resolving vectors

Displacement and velocity

Acceleration and velocity-time graphs

Equations of motion

Acceleration and free fall

Projectile motion

Mass, weight and centre of mass

Resolving forces

Triangle of forces

The principle of moments

Working as a physicist

Physical quantities and units

How to plan an experiment

Hazards, risks and results

Analysing data

Evaluating data

Explainer Video

Tutor: Lex

Summary

Resolving vectors

In a nutshell

Resolving vectors is the process of turning a vector into two perpendicular components, which allows vectors to be added together when they are at any angle to each other.

Definitions

Key word	Definition
Vector	A quantity that has both a magnitude and a direction
Resolving a vector	Splitting a vector into two perpendicular components

Equations

description	equation
Horizontal component of a vector	$F_x = F\cos\theta$
Vertical component of a vector	$F_y = F\sin\theta$

Resolving a vector

The two components of any vector is given by the following two equations.

$F_x = F\cos\theta$

$F_y = F\sin\theta$

where $F_x$ and $F_y$ are the horizontal and vertical components of the vector respectively, $F$ is the magnitude of the vector and $\theta$ is the angle the vector makes with the horizontal.

Physics; Motion; KS5 Year 12; Resolving vectors

1.	Hypotenuse - $F$
2.	Opposite - $F_y$
3.	Adjacent - $F_x$

This graphic shows the locations of $F_x$ , $F_y$ and $F$ on a triangle. The trigonometric ratios relating the lengths and angles of a triangle are as follows.

$\sin\theta = \dfrac{opposite}{hypotenuse}$

$\cos\theta = \dfrac{adjacent}{hypotenuse}$

$\tan\theta = \dfrac{opposite}{adjacent}$

Substituting in $F_x$ , $F_y$ and $F$ for the adjacent, opposite and hypotenuse of a triangle then rearranging the equations will give the equations for the components of a vector.

Example

A force of magnitude $10N$ acts at $60\degree$ to the horizontal. Calculate the horizontal and vertical components.

First, write down the equations.

$F_x=F\cos\theta$

$F_y=F\sin\theta$ 

Next, substitute in the values.

$F_x = 10 \times \cos60$

$F_x = 5N$ 

$F_y=F\sin\theta$ 

$F_y = 8.66N$ 

The horizontal component is $\underline{5N}$ and the vertical component is $\underline{8.66N}$ .

Adding non-perpendicular vectors

In order to add vectors that are not perpendicular to each other, the vectors must first be resolved into their components. Then, the total magnitudes in the horizontal and vertical directions are calculated. Finally, the components can be combined using Pythagoras' theorem to find the final magnitude and direction of the resultant.

Example

Calculate the resultant of the following two forces.

First, write down the equations to resolve the forces.

$F_x = F\cos\theta$ 

$F_y = F\sin\theta$ 

Next, substitute in the values.

$F_x = 10\times\cos45$ 

$F_x = 7.1N$ 

$F_y = 10\times\sin45$ 

$F_y = 7.1N$ 

Next, sum the horizontal and vertical forces.

$R_x = 6 + 7.07$ 

$R_x = 13.1N$ 

$R_y = 7.1N$ 

Next, use Pythagoras' theorem to calculate the resultant force.

$R = \sqrt{7.1^2 + 13.1^2}$ 

$R = 14.9N$ 

Finally, use trigonometry to calculate the resultant force's direction.

$\tan\theta = \dfrac{opp}{adj}$ 

$\tan\theta = \dfrac{7.07}{13.07}$ 

$\theta = 28.4\degree$ 

The resultant force has a magnitude of $\underline{14.9N}$ and a direction of $\underline{28.4\degree}$ .

Learn with Basics

Length:

Unit 1

Resolving forces

Unit 2

Resolving forces

Jump Ahead

Optional

Unit 3