Scalars and Vectors

(... and Matrices)

What are Scalars and Vectors?

A scalar has only magnitude (size):

3.044, −7 and 2½ are scalars

Distance, speed, time, temperature, mass, length, area, volume, density, charge, pressure, energy, work and power are all scalars.

A vector has magnitude and direction:

vector magnitude and direction

Displacement, velocity, acceleration, force and momentum are all vectors.

And watch out for these special words:

displacement vs distance

Distance vs Displacement

You can walk a long distance, but your displacement may be small (or zero if you return to the start).

Speed vs Velocity

dog run ball

Saying Ariel the Dog runs at 9 km/h (kilometers per hour) is a speed.

But saying he runs 9 km/h Westwards is a velocity.

See Speed and Velocity to learn more.


A vector is often written in bold, like a or b so we know it is not a scalar:

Example: kb is actually the scalar k times the vector b.

vector notation

A vector can also be written as the letters of its head and tail with an arrow above it, like this:

Using Scalars

Scalars are easy to use. Just treat them as normal numbers.

Example: 3 kg + 4 kg = 7 kg

Using Vectors

The page on vectors has more detail, but here is a quick summary:

We can add two vectors by joining them head-to-tail:

vector addition

We can subtract one vector from another:

vector subtraction

We can multiply a vector by a scalar (called "scaling" a vector):

Example: multiply the vector m = (7,3) by the scalar 3

vector scaling   a = 3m = (3×7,3×3) = (21,9)

It still points in the same direction, but is 3 times longer

(And now you know why numbers are called "scalars", because they "scale" the vector up or down.)

Polar or Cartesian

A vector can be in:

Like this:

vector polar <=> vector cartesian
Vector a in Polar
  Vector a in Cartesian

(Read how to convert them at Polar and Cartesian Coordinates.)

Example: the vector 13 at 22.6°

In Polar (magnitude and direction) form:

coordinates polar 13 at 22.6 degrees
The vector 13 at 22.6°

Is approximately (12,5) In Cartesian (x,y) form:

coordinates polar 13 at 22.6 degrees
The vector (12,5)

Have a try of the Vector Calculator to get a feel for how it all works.

Multiplying a Vector by a Vector (Dot Product and Cross Product)

dot product 1

How do we multiply two vectors together? There is more than one way!

(Read those pages for more details.)

More Than 2 Dimensions

Vectors also work perfectly well in 3 or more dimensions:

vector 3d: (1,4,5)
The vector (1,4,5)

List of Numbers

So a vector can be thought of as a list numbers:


Scalars, Vectors and Matrices

And when we include matrices we get this interesting pattern:

scalar vector matrix

In fact a vector is also a matrix! Because a matrix can have just one row or one column.

So the rules that work for matrices also work for vectors.