A force of 10 N to the right and 10 N to the left cancel out — net force is zero.
If force were scalar (no direction), you couldn't distinguish these.
VECTOR REPRESENTATION:
An arrow represents a vector.
Length of arrow ∝ magnitude.
Direction of arrow = direction of the quantity.
Distinguishing Common Pairs
Several pairs are commonly confused:
DISTANCE vs DISPLACEMENT:
Distance: total path length travelled — scalar.
Displacement: straight-line distance from start to finish, with direction — vector.
Example: walking 3 m east then 3 m west → distance = 6 m; displacement = 0 m.
SPEED vs VELOCITY:
Speed: how fast — scalar (magnitude only).
Velocity: speed in a specified direction — vector.
Example: car at 30 m/s — speed = 30 m/s. Velocity = 30 m/s north.
MASS vs WEIGHT:
Mass: amount of matter — scalar (kg).
Weight: gravitational force — vector (N, acts downward).
Adding Vectors
When adding SCALAR quantities: simple arithmetic.
3 kg + 5 kg = 8 kg (always).
When adding VECTOR quantities: direction matters.
Two forces in the SAME direction: add magnitudes.
10 N + 5 N (both right) = 15 N right.
Two forces in OPPOSITE directions: subtract smaller from larger.
10 N right + 5 N left = 5 N right (resultant).
Two forces at RIGHT ANGLES: use Pythagoras.
Resultant² = F₁² + F₂²
3 N up + 4 N right → resultant = √(9 + 16) = 5 N (at an angle).
Scale drawings can also be used to find the resultant of vectors at any angle.
⚠️ Common Mistake
SPEED is scalar, VELOCITY is vector. DISTANCE is scalar, DISPLACEMENT is vector. The most common error is treating velocity and speed as the same thing. A car travelling in a circle at constant SPEED has changing VELOCITY (direction changes).