# Vectors

A vector is a quantity with both **magnitude and direction, **and is typically represented visually by a line segment between two points.

There are many ways of representing vectors in notation:

The two points that the vector connects with an

**arrow above them**As a

**bold typeface**lower case letterAs an

__underlined lower case letter__As a

**column vector**, showing displacement in the x-direction above that in the y-directionAs a multiple of

**unit vectors**,**i**(one unit in the positive x-direction) and**j**(one unit in the positive y-direction)

These notes will predominantly use **bold typeface** of lower case letters, __underlined__ letters and the unit vectors **i** and **j**.

Vectors can be **multiplied by a scalar, and added and subtracted:**

## Magnitude & Direction

The **magnitude **of a vector is given by Pythagoras' Theorem. Magnitude is noted using straight lines on either side of the letter, like modulus.

For the vector=axi+yj,|= √(a|x²+y²)

A** unit direction vector, â**, can be found as **a** / **|a|**

A vector can also be defined by giving its magnitude and the angle it makes with one of the coordinate axis. This is called **magnitude-direction form**.

## Position Vectors

**Position vectors** are used to give the location of a point relative to a fixed origin.

A point (p,q) has a position vector pi + qj

### Vector Geometry

Position vectors can be used to solve geometric problems:

## Vectors in 3D

In three-dimensional geometry, there are x, y, and z-axes. This means each coordinate has three values, (x, y, z). The position vector of this is x**i** + y**j** + z**k**

Pythagoras' Theorem still applies, but adjusted for three points:

Distance from point (x, y, z) to origin is given as √(x²+y²+z²)

The distance between two points, (x₁, y₁, z₁) and (x₂, y₂, z₂) is given as:

√( (x₁ -x₂)² +(y₁ -y₂)² + (z₁ -z₂)² )