Imagine you're telling a friend about your day. You might say, "It was 35 degrees outside." That's a complete piece of information — just a number with a unit. Now imagine saying, "I walked 5 kilometres." That's also a number with a unit, but something feels incomplete. Where did you walk? Did you go north? East? In circles? The "5 kilometres" alone doesn't tell the full story.
This difference is the entire idea behind scalars and vectors.
The Intuition
Scalars are quantities that are fully described by a magnitude (a number and a unit). Temperature, mass, time, speed, energy — these are scalars. If I say "the mass is 10 kg," you know everything there is to know about that mass. There's no direction to mass.
Vectors are quantities that need both magnitude and direction to be fully described. Displacement, velocity, force, acceleration — these are vectors. "5 km north" is a vector. "10 N downward" is a vector. The direction is not optional; it's part of the quantity itself.
Note
Speed is a scalar (just "how fast"). Velocity is a vector ("how fast" + "in which direction"). A car moving at 60 km/h has a speed of 60 km/h. If it's moving east at 60 km/h, its velocity is 60 km/h east.
The Precise Statement
A scalar is a physical quantity that has only magnitude. It obeys ordinary arithmetic: you can add, subtract, multiply, and divide scalars just like numbers.
A vector is a physical quantity that has both magnitude and direction. Vectors obey special rules of addition (like the triangle law or parallelogram law) because direction matters.
A vector is often represented as A or in bold as A. Its magnitude is written as ∣A∣ or simply A.
Key Differences at a Glance
Property
Scalar
Vector
Description
Magnitude only
Magnitude + Direction
Example
25°C, 10 kg, 5 seconds
10 m/s north, 20 N downward
Addition
Ordinary arithmetic
Triangle/parallelogram law
Division by a scalar
Yes (e.g., distance ÷ time = speed)
Yes (e.g., displacement ÷ time = velocity)
Division by a vector
Not defined
Not defined
Why This Matters
When you add two scalars, say 5 kg + 3 kg, you get 8 kg. Simple.
When you add two vectors, say walking 3 km east then 4 km north, you don't get 7 km. You get 5 km northeast (by Pythagoras). The direction of each step matters. If you walked 3 km east and then 4 km west, you'd end up 1 km east — not 7 km anywhere.
Watch out
A common mistake is treating vectors like scalars in addition. Never simply add magnitudes of vectors unless they point in exactly the same direction. Always account for direction.
The Mathematical Representation
In one dimension, we often use a sign to indicate direction: +5 m/s means right, −5 m/s means left. The sign is the direction. …
A vector is a physical quantity possessing both magnitude and direction, and it must combine with other vectors according to the triangle/parallelogram law of vector addition.
Many physical quantities need more than just a number to describe them fully — they also need a direction. Such quantities are called vectors.
A vector quantity is defined as one that has:
Magnitude (a numerical value with units), and
Direction, and
It obeys the laws of vector addition (triangle law / parallelogram law) — simply having a direction is not enough; the quantity must add the way a directed line segment does.
Vectors are typically represented by an arrow: the length of the arrow shows the magnitude and the arrowhead shows the direction.