Matrix Cycle in 3D

Linear transformation in 3D graphics can be tricky, here is as “Matrix Cycle” that should help to better understand transition between spaces. The tricky part is actual numerical representation (or direction) in the transformation matrix.   Some info about  also here (nice notation of actual numerical matrix direction) nice online demo here 

following picture is from :http://antongerdelan.net/opengl/raycasting.html

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Geometric proof of vector dot product

dotProof Source: colorful introduction to Linear Algebra (may 2014) Cosine Similarity – understanding it as volume of parallelepiped cosineSimilarity Another good source for understanding dot product : http://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/ dot_product_components dot_product_rotation

other resources:

Significance &Application of Cross product and Dot product. http://visualizingmathsandphysics.blogspot.cz/2013/06/vectors-significance-of-cross-product.html

Independence of  Perpendicular components of the motion http://www.physicsclassroom.com/Class/vectors/u3l1g.cfm

Math Insight: http://mathinsight.org/dot_product

\frac{a\cdot b}{\left \| a\right \|}= cos\alpha \left \| b \right \|

“projection” of vector b on a equals cosine of angle between them times the length of b or vice versa projection of a on b:

dot_product_projection

Dot product Identities, some pictures taken from http://gamemath.com/

dotProductIdentities2