# 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**:

Displacement, velocity, acceleration, force and momentum are all vectors.And watch out for these special words:

### Distance vs Displacement

- Distance is a scalar ("3 km")
- Displacement is a vector ("3 km Southeast")

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

### Speed vs Velocity

- Speed is how fast something moves.
- Velocity is speed with a
**direction**.

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.

## Notation

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

- so
**c**is a vector, it has magnitude and direction - but c is a scalar, like 3 or 12.4

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

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:

We can subtract one vector from another:

- first we reverse the direction of the vector we want to subtract,
- then add them as usual:

**a** − **b**

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

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

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:

**magnitude and direction**(Polar) form,- or in
**x and y**(Cartesian) form

Like this:

<=> | ||

Vector a in PolarCoordinates | Vector a in CartesianCoordinates |

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

### Example: the vector **13 at 22.6°**

In Polar (magnitude and direction) form:

The vector **13 at 22.6°**

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

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)

How do we - The scalar or Dot Product (the result is a scalar).
- The vector or Cross Product (the result is a vector).
(Read those pages for more details.) |

## More Than 2 Dimensions

Vectors also work perfectly well in 3 or more dimensions:

**The vector (1,4,5)**

## List of Numbers

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

- 2 numbers for 2D space, such as (4,7)
- 3 numbers for 3D space, such as (1,4,5)
- etc

## Scalars, Vectors and Matrices

And when we include matrices we get this interesting pattern:

- A
**scalar**is a number, like**3, -5, 0.368, etc**, - A
**vector**is a**list**of numbers (can be in a row or column), - A
**matrix**is an**array**of numbers (one or more rows, one or more columns).

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.