[Guide] Quick and dirty Introduction to Vector Math

A Vector, in mathematical terms, is a multi-dimensional 'step.' That is, it is a step (or, difference) in multiple directions at once. As far as SE (and physical reality) is concerned there are 3 Dimensions, typically expressed as X, Y and Z co-ordinates.

A Vector, in programming terms, is a construct that can hold the X, Y and Z measurements. There are two ways to use these measurements.

1. Positional: Measures offset from some other position
2. Directional: Measures a unit direction

This can be a bit confusing at first. The key point to remember is that when using a Vector as Directional, it must have a length as close to 1 as possible. This is called Normalising, and is easy to do

Code:

 var vecUnitLength = Vector3.Normalize( vecTotalDisplacement ); 

Some standard Space Engineers Directional vectors:

Code:

[FONT= &#39]var Forward = new Vector3( 0, 0,-1);[/FONT]
[FONT= &#39]var Back    = new Vector3( 0, 0, 1);[/FONT]
[FONT= &#39]var Left    = new Vector3(-1, 0, 0);[/FONT]
[FONT= &#39]var Right   = new Vector3( 1, 0, 0);[/FONT]
[FONT= &#39]var Up      = new Vector3( 0, 1, 0);[/FONT]
[FONT= &#39]var Down    = new Vector3( 0,-1, 0);[/FONT]

Get the components using

Code:

 vec.GetDim(n); // n: 0 for X, 1 for Y and 2 for Z 

Use string formatting when viewing

Code:

 vec.GetDim(n).ToString("N2"); // Number, two decimal places

To get a displacement to vecA from vecB

Code:

 var vecBA = vecA - vecB; 

To get a direction to vecA from vecB

Code:

 var vecDirBA = Vector3.Normalize( vecA - vecB ); 

To get an angle requires two Vectors

Code:

 var angle = Math.Acos( vec1.Dot( vec2 ) ); 

To get the axis of rotation

Code:

 var axis = vec1.Cross( vec2 ); 

To rotate, or Transform, a Vector between reference frames (or, co-ordinate spaces) we require a Transformation Matrix.
To create a Matrix for a rotation

Code:

 var matTransform = Matrix.CreateFromAxisAngle( axis, angle ); 

To perform the vector transform

Code:

 var vecTransformed = Vector3.Transform( vecOriginal, matTransform );

 

Code:

var angle = Math.Acos( vec1.Dot( vec2 ) ); 

assumes vec1 and vec2 are normalized.

more generally it would be:

Code:

var angle = Math.Acos( vec1.Normalize().Dot( vec2.Normalize() ) ); 

or alternatively:

Code:

var angle = Math.Acos( vec1.Dot( vec2 )  / (vec1.Length()*vec2.Length()));