3.4.1.1MechanicsFoundation

Scalars and Vectors

Every quantity in physics is either a $scalar$ — described by size alone — or a $vector$ — described by size $and$ direction. Getting this distinction right is the foundation of all of mechanics, because vectors must be combined geometrically, not just added up like ordinary numbers.

30 min Video by Science Shorts 3.4.1 Force, energy and momentum

Vectors — AS/A-Level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

State the difference between a scalar and a vector quantity
Classify common quantities (e.g. mass, force, velocity) as scalar or vector
Add two vectors using the tip-to-tail method and find the resultant
Find the resultant of two perpendicular vectors using Pythagoras and trigonometry
Resolve a single vector into two perpendicular components

Scalars vs vectors

A $scalar$ has magnitude (size) only. A $vector$ has magnitude $and$ a direction. For example, a speed of $30$ m/s is a scalar, but a velocity of $30$ m/s due north is a vector.

The classic trap is distance vs displacement and speed vs velocity: the first of each pair is a scalar, the second is a vector that also records direction.

Tip — Learn the standard lists. Scalars: distance, speed, mass, energy, time, temperature. Vectors: displacement, velocity, acceleration, force, momentum.

Adding vectors

To add vectors you place them $tip-to-tail$ : draw the first, then start the second where the first ends. The $resultant$ is the single vector from the very start to the very end.

When two vectors act at right angles, the resultant is the hypotenuse of a right-angled triangle, so its magnitude comes from Pythagoras and its direction from trigonometry.

R = F_{x}^{2} + F_{y}^{2}, tan θ = \frac{F _{y}}{F _{x}}

Resultant magnitude and direction for two perpendicular vectors

F_{x}

and

F_{y}

1Magnitude:

R = 3. 0^{2} + 4. 0^{2} = 25 = 5.0

2Direction:

tan θ = \frac{4.0}{3.0}

, so

θ = 5 3^{\circ}

north of east.

Answer

R = 5.0

N at

5 3^{\circ}

north of east

Resolving vectors into components

The reverse of adding is $resolving$ : splitting one vector into two perpendicular components, usually horizontal and vertical. A vector of magnitude $F$ at angle $θ$ to the horizontal has the components below.

This is the single most useful skill in mechanics — it lets you handle forces on slopes, projectiles, and equilibrium problems one direction at a time.

F_{x} = F cos θ

Component

adjacent

to the angle uses cosine.

F_{y} = F sin θ

Component

opposite

the angle uses sine.

1Horizontal:

F_{x} = 20 cos 3 0^{\circ} = 17.3

2Vertical:

F_{y} = 20 sin 3 0^{\circ} = 10

Answer

F_{x} = 17.3

F_{y} = 10

Tip — Whether a component uses sin or cos depends on where the angle is measured from, not on a fixed rule. The component next to (adjacent to) the angle always uses cosine.

Equation recap

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

Resultant of two perpendicular vectors.

F_{x} = F cos θ

Horizontal component (adjacent to angle).

F_{y} = F sin θ

Vertical component (opposite the angle).

Common mistakes to avoid

Adding vectors arithmetically, e.g. 3 N east + 4 N north = 7 N.

Combine them geometrically: the resultant is 5 N (Pythagoras), not 7 N.

Always using cosine for the horizontal component.

Use cosine for the component adjacent to the given angle — which one that is depends on how the angle is defined.

Calling speed a vector.

Speed is a scalar; velocity is the vector (speed in a stated direction).

Key takeaways

Scalars have magnitude only; vectors have magnitude and direction.
Add vectors tip-to-tail; the resultant runs from the start of the first to the end of the last.
For perpendicular vectors, use Pythagoras for size and tan for direction.
Resolve a vector with F cosθ (adjacent) and F sinθ (opposite).

Test yourself

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