2.1.2FoundationsFoundation

Scalars and Vectors

Say a car travels "at 20 m/s" and you’ve told half the story — 20 m/s in which direction? Some physical quantities need only a size to be completely described; others are meaningless without a direction too. Sorting quantities into these two camps — $scalars$ and $vectors$ — and learning to combine and split vectors correctly is a skill used in essentially every module of this course.

30 min Video by Science Shorts Module 2.1 Physical quantities and units

Easy Vector Resolving Trick — GCSE & A-level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

Distinguish scalar quantities from vector quantities, with examples of each
Add two vectors graphically and by calculation
Resolve a single vector into two perpendicular components
Find the resultant of several vectors using components
Apply vector resolution to a force acting at an angle

Scalars and vectors

A $scalar$ quantity has magnitude (size) only — mass, energy, speed, distance, time and temperature are all scalars. A $vector$ quantity has both magnitude and direction — displacement, velocity, acceleration, force and momentum are all vectors.

The distinction matters because vectors combine differently from ordinary numbers: two forces of 3 N and 4 N acting on an object do not necessarily add to 7 N — the answer depends entirely on the angle between them.

Tip — A quick test: if the quantity could sensibly be negative or point "the other way", it’s a vector. Speed (always positive, no direction) is a scalar; velocity (same size, but with direction) is the corresponding vector.

Combining vectors

Vectors are combined "tip-to-tail": draw the first vector, then start the second from the tip (arrowhead) of the first; the $resultant$ is the single vector from the start of the first to the tip of the second. When two vectors are perpendicular, the resultant’s magnitude follows directly from Pythagoras, and its direction from basic trigonometry.

R = F_{x}^{2} + F_{y}^{2} θ = tan^{- 1} (\frac{F _{y}}{F _{x}})

Resultant of two perpendicular vectors, and its angle from the Fx axis.

1These forces are perpendicular, so use Pythagoras:

R = 40 0^{2} + 30 0^{2} = 160000 + 90000

R = 250000

3Direction:

θ = tan^{- 1} (\frac{300}{400}) = tan^{- 1} (0.75)

Answer

R = 500

N, at

36.9°

from the forward direction.

Resolving a vector into components

The reverse process — splitting one vector into two perpendicular parts — is $resolving$ it, and is often the more useful skill in practice. A vector $F$ acting at angle $θ$ to a chosen axis has a component along that axis and a component perpendicular to it, found using basic trigonometry (SOHCAHTOA).

F_{x} = F cos θ F_{y} = F sin θ

Components of a vector F at angle θ to the x-axis.

1Horizontal component:

F_{x} = F cos θ = 80 cos 30°

F_{x} = 80 \times 0.866

3Vertical component:

F_{y} = F sin θ = 80 sin 30° = 80 \times 0.5

Answer

F_{x} \approx 69.3

N (horizontal),

F_{y} = 40

N (vertical).

Tip — Resolving into components is the single most useful trick for handling more than two vectors at once — add all the x-components together, add all the y-components together, then combine those two totals with Pythagoras at the very end.

Equation recap

F_{x} = F cos θ, F_{y} = F sin θ

Resolving a vector into perpendicular components.

R = F_{x}^{2} + F_{y}^{2}

Magnitude of the resultant of perpendicular components.

Common mistakes to avoid

Adding two vectors as if they were plain numbers, ignoring direction.

Vectors only add directly if they point in exactly the same direction — otherwise use components or a scale diagram to find the true resultant.

Mixing up sine and cosine when resolving a vector.

The component ALONG the angle uses cosine; the component perpendicular to it uses sine — sketch the angle and check which side is adjacent before choosing.

Forgetting to resolve every vector before summing when combining three or more forces.

With more than two vectors, resolve ALL of them into x and y components first, sum each set separately, then combine the two totals at the end.

Key takeaways

Scalars have magnitude only (mass, energy, speed); vectors have magnitude and direction (velocity, force, momentum).
Perpendicular vectors combine via Pythagoras for magnitude and tan⁻¹ for direction.
A vector at angle θ resolves into components Fcosθ (along the axis) and Fsinθ (perpendicular to it).
To combine several vectors, resolve each into components, sum each set, then find the resultant magnitude and direction.

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Scalars and Vectors

scalars

and

vectors

— and learning to combine and split vectors correctly is a skill used in essentially every module of this course.

30 min Video by Science Shorts Module 2.1 Physical quantities and units