Physics deals with two kinds of quantities — scalars and vectors. The distinction is simple, but its consequences are deep: from notation, to how you add them, to their physical meaning.
Scalars — a quantity with magnitude only
A scalar is just a number, no direction. Examples:
| Quantity | Sample value | Units |
|---|---|---|
| Mass | 2.5 |
kg |
| Temperature | −15 |
°C |
| Time | 10.3 |
s |
| Energy | 500 |
J |
| Electric charge | 1.6 × 10⁻¹⁹ |
C |
| Density | 1000 |
kg/m³ |
Scalars combine by ordinary arithmetic: add, subtract, multiply, divide. A 5 kg rock + a 3 kg rock = 8 kg. Done.
Vectors — a quantity with magnitude AND direction
A vector carries two pieces of information: a magnitude (how much) and a direction (which way). Examples:
| Quantity | Sample |
|---|---|
| Displacement | 10 m north |
| Velocity | 20 m/s east |
| Acceleration | 9.8 m/s² downward |
| Force | 50 N horizontal |
| Electric field | 100 N/C in +x direction |
Note: without direction these are incomplete. “A 50-N force” says nothing on its own — right, left, up all give different physics.
Notation
Vectors are marked with one of:
- Arrow over the letter:
\vec a,\vec F,\vec v— common in handwriting and prose - Bold letter:
\mathbf{a},\mathbf{F},\mathbf{v}— common in print
The magnitude is written with |·| or the plain letter:
$$
|\vec F| = F
$$
So F (no arrow) means the magnitude; \vec F is the vector.
On this site: we mostly use arrows for clarity: \vec a.
A vector on a diagram
A vector is drawn as an arrow:
– length = magnitude (any convenient scale)
– direction = pointed by the arrow
– tail and head — head indicates direction
Two vectors are equal — precise definition
Two vectors are equal if:
– their magnitudes match
– their directions match
Important: the position of a vector doesn’t matter. Slide \vec a parallel to itself and it remains the same vector — magnitude and direction are unchanged. This freedom is powerful: for addition, you can move vectors around to make the geometry convenient.
The opposite vector
-\vec a means a vector with the same magnitude but opposite direction. On a diagram, rotate the arrow 180°.
$$
\vec a + (-\vec a) = \vec 0
$$
where \vec 0 is the zero vector — magnitude zero, direction undefined.
Vectors in different dimensions
- 1D: a vector reduces to a signed number (
+or-). That was Chapter 2 — velocity +20 to the positive direction, velocity −20 to the negative. - 2D: a vector in a plane — two components (
x,y). - 3D: a vector in space — three components (
x,y,z). - 4D or more: abstract vectors — in relativity (4D spacetime), in quantum mechanics (infinite-dimensional Hilbert space).
In this chapter we focus on 2D and 3D. Learn the concepts once in 2D and extending to 3D is easy.
A few common mistakes
1. “A vector is just a directed line” — not quite.
A straight line is a geometric shape; a vector connects two points with a direction. A line without direction isn’t a vector.
2. Don’t confuse scalar and vector.
Temperature is a scalar. Temperature gradient is a vector (toward increasing T). Speed is a scalar but velocity vector is a vector.
3. Magnitude is always non-negative.
|\vec a| \geq 0 always. Saying “the magnitude of this vector is negative” is wrong — you probably confused a component with the magnitude.
4. Only magnitude isn’t full information.
“The car is doing 60 km/h” — which way? Without direction that’s half a story.
What you should be able to do
After this section, you should be able to:
- Explain the difference between scalar and vector with examples
- Name 5 scalar and 5 vector quantities in physics
- Recognize vector notation (arrow or bold)
- Draw a vector on a diagram with tail, head, length
- Define the opposite vector and zero vector
- Explain why a vector’s position doesn’t matter — only its magnitude and direction
Preview of §3.2
Now that we have the definition, next question: how do we add two vectors? Answer: scalar addition doesn’t work — the directions may differ. §3.2 introduces the graphical (tip-to-tail) method. §3.3 introduces the algebraic (component) method.
📚 See also: Halliday Vol 1, Ch 3, §3.1 — Scalars and Vectors.
📖 Open reference: OpenStax University Physics Vol 1 — Chapter 2.1: Scalars and Vectors.
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