3.1 — Scalars and vectors

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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