Game Creation with XNA/Mathematics Physics/Vectors and Matrices

Vectors and Matrices

We need to recall some basic facts about vector and matrix algebra, especially when trying to develop 3D games. A nice introduction with XNA examples can be found in the book by Cawood and McGee.

A right triangle showing the relation between opposite, adjacent and hypotenuse.

Right Triangle

Once upon a time there was little Hypotenuse. He had two cousins: the Opposite and his sister the Adjacent. Both were usually just known by their nick names 'Sine' and 'Cosine'. They lived together in a right triangle close to the woods. They were related through his mother's sister, aunty Alpha. His father, who was a mathematician, used to say that:

$\sin \alpha ={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {a}{h}}$

$\cos \alpha ={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}={\frac {b}{h}}.$

Sometimes he also referred to uncle Tangent (who was married to aunty Alpha) and said that

$\tan \alpha ={\frac {\textrm {opposite}}{\textrm {adjacent}}}={\frac {a}{b}},$

so in a sense uncle Tangent of aunty Alpha was Sine divided by Cosine. To us that didn't make any sense, but Hypotenuse's father said that was how it always was.