Following on from the post on Monday, today I will be talking about the basic properties of vectors.

Vectors have a magnitude (size) and direction:

They are normally depicted using a bold alphabetical letter, e.g. **a**, or can be written as the letter of its initial and terminal position with an arrow above it.

**Addition and Subtraction**

The sum of any vectors **a** and **b** is:

**a** + **b** = (a1 + b1)**i** + (a2 + b2)**j** + (a3 + b3)**k**

This can be represented geometrically by placing the tail of the arrow **b** at the head of arrow **b**. The arrow that can be drawn from the tail of **a** to the head of **b **is the resultant vector – **a** + **b**.

This is sometimes referred to as the parallelogram rule as **a** and **b** form the sides of a parallelogram with **a** + **b **as one of its diagonals.

The subtraction of two vectors **a** and **b** is:

**a** – **b** = (a1 – b1)**i** + (a2 – b2)**j** + (a3 – b3)**k**

This can be represented geometrically as the line that connect the tail of **b** with the tail of **a**.

**Length/Magnitude**

This is denoted by **||a||** or **|a|** (do not confuse this with the absolute value!). The length of a vector can be calculated using Pythagoras’ theorem since the unit vectors ** i, j **and

*are orthogonal:*

**k****Unit Vector**

A unit vector is any vector that has a magnitude of 1, and are therefore is used to indicate the direction. A unit vector, normally indicated with a hat **â,** can be calculated by ‘normalising’ a vector, that is, divide the vector by its magnitude:

**Dot Product**

The dot product of two vectors is denoted by **a.b** and is defined by:

where θ is the actue angle between **a** and **b. **

The dot product can also be calculated by adding the products of the components of each vector:

Geometrically, this can be described as drawing **a** and **b** with a common start point, and then multiplying the length of **a** with the length of that component of **b** that points in the same direction as **a**.

**Cross Product**

The cross product is only meaningful in 3 (or 7) dimensions and its result is a vector (unlike that of the dot product).

The cross product is denoted by **a x b **and is defined as being the vector perpendicular to both **a** and **b**. It can be calculated using the following equations:

where **n** is a unit vector perpendicular to both **a** and **b;**

or

The ‘right-hand’ rule is used to see which direction the cross product points, as it could point in the completely opposite direction:

Source: 1

I must admit that vectors are not by strongest area in mathematics, so it’s really interesting for me to do some research on them myself! Let me know what you think. M x

## One thought on “Vectors #2: The Basics”