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Department of Architecture ARCH 655 Parametric Modeling in Design Vector Mathematics: Applications Wei Yan, Ph.D. Associate Professor Department of Architecture Texas A&M University

Use of Dot Product

Extract the projection The dot product between the unit x vector (1,0,0) and a vector V gives the component of V in the x-direction.

Work out the angle between 2 vectors cos(theta) = a.b / (|a||b|) = (a1*b1 + a2*b2 + a3*b3) / (sqrt(a12 + a22 +a32) * sqrt(b12 + b22 +b32))

The Dot Product measures the 'Right-Angularity' of 2 vectors Vectors a and b are orthogonal if, and only if, a.b = 0.

Use of Cross Product

Cross product of 2 vectors, results in 3rd vector which is perpendicular to both of them. For finding surface normal

|a x b| = |a||b|sin(theta) = base * height The length of the cross product is the area of the parallelogram made by the two vectors.

Use of vectors Many problems can be solved most easily by using vector operations. (And some problems can't really be expressed very well, without reference to vector operations.) Geometry: find normal, find average direction of two lines, find projection, … Physics: velocity, force http://www.mathrec.org/vector.html

General Tools http://www.wolframalpha.com/input/?i=vector+%281%2C+2%2C+6%29+%2B+%284%2C+5%2C+3%29