Scalars and Vectors: Basic Definitions

A scalar is a physical quantity that has only magnitude (size), such as mass, time or temperature, while a vector is a quantity that has both magnitude and direction, such as displacement, velocity and force. In brief: scalars add like ordinary numbers, but vectors add geometrically, taking direction into account. Vectors are drawn as arrows, with length proportional to magnitude. Distance and speed are scalars; displacement and velocity are their vector counterparts.

If you are told that a ship traveled 20 km, you do not know where it ended up. If you are told it traveled 20 km north, you do. Directions matter for many quantities in physics. This guide defines scalars and vectors, shows how vectors are represented and added, compares scalar and vector quantities in a table, works through an example of displacement and distance and corrects common mistakes.

What are scalars?

A scalar is completely described by a number and a unit. Examples: mass (3.0 kg), time (12 s), temperature (20 °C), distance (50 m), speed (8 m/s), energy (100 J), density and volume. Scalars are combined by ordinary arithmetic: 2 kg + 3 kg = 5 kg.

What are vectors?

A vector needs a magnitude, a unit and a direction. Examples are displacement (5 m east), velocity (10 m/s north), acceleration, force (20 N downward), weight and momentum.

• Representation: an arrow whose length is proportional to the magnitude, and which points in the direction of the vector. A vector is written in bold (v) or with an arrow over the letter, and its magnitude is written v or |v|.
• Direction is given by compass directions (north, east), by an angle from a reference line (30° above the horizontal), or by signs in one dimension (positive to the right, negative to the left).
• Equal vectors have the same magnitude and direction, wherever they are drawn. The negative of a vector has the same magnitude and the opposite direction.
• Components: a vector in a plane can be split into two perpendicular parts, usually horizontal and vertical. For a vector of magnitude v at angle θ above the horizontal, the components are v cos θ and v sin θ.

How are vectors added?

The sum of vectors is called the resultant. For vectors in the same line, add magnitudes if they point the same way, and subtract if they point in opposite ways. For vectors at an angle, use the head-to-tail method: draw the first vector, then draw the second starting at the head of the first. The resultant goes from the tail of the first to the head of the last. For perpendicular vectors, the magnitude is found with the Pythagorean theorem, R = √(a² + b²), and the direction with tan θ = b ÷ a. The parallelogram method gives the same result. Multiplying a vector by a positive scalar changes its magnitude but not direction, and multiplying by a negative scalar reverses it.

Scalar and vector quantities compared

Worked example: distance and displacement

A student walks 3.0 m east, then 4.0 m north, in a total time of 10 s.

1. Distance (scalar): 3.0 m + 4.0 m = 7.0 m.
2. Displacement (vector): the two legs are perpendicular. Magnitude R = √(3.0² + 4.0²) = √25 = 5.0 m.
3. Direction: tan θ = 4.0 ÷ 3.0 = 1.33, so θ = 53° north of east.
4. Average speed: 7.0 m ÷ 10 s = 0.70 m/s (scalar).
5. Average velocity: 5.0 m ÷ 10 s = 0.50 m/s, 53° north of east (vector).

The distance (7.0 m) is greater than the magnitude of the displacement (5.0 m). If the student walked back to the start, the displacement would be zero, but the distance would be 14.0 m.

How were vectors developed?

The parallelogram rule for forces was used by Simon Stevin around 1600 and by Isaac Newton in the Principia. The modern vector notation was developed in the late 1800s by Josiah Willard Gibbs and Oliver Heaviside, who simplified William Rowan Hamilton’s quaternions, and it became standard in physics because it describes electric and magnetic fields conveniently.

Which mistakes are common?

Mistake 1: Adding vectors like numbers

3 N and 4 N do not always give 7 N. The resultant is 7 N only when both act in the same direction, 1 N when opposite, and 5 N when perpendicular.

Mistake 2: Using speed and velocity as the same quantity

Speed has no direction, and velocity does. A car going around a circle at constant speed has changing velocity.

Mistake 3: Treating a negative sign as less magnitude

In one dimension, the sign only shows the direction. −5 m/s has the same speed as +5 m/s.

Mistake 4: Forgetting to state the direction

An answer for a vector is incomplete without a direction.