Vector Products

Definitions of Terms

Vector Components: Given a vector, a, with its initial point at (xi , yi , zi) and terminal point at (xt , yt , zt) , the components are :

xa = xt- xi
ya = yt- yi
za = zt- zi

Unit Vectors: The components of a vector are multiplied by unit vectors, assigning directions to the components with respect to the x, y and z axes. The unit vectors are defined as :

i = (1, 0, 0)
j = (0, 1, 0)
k = (0, 0, 1)

Magnitude: |a| = The square root of  xa2 + ya2 + za2

Angle: (a,b) means the angle between the vectors a and b

Cross Product: Defined as a third vector, c = |a| |b| sin (a,b)
The direction of c is perpendicular to the plane of a and b, and follows the advance of a right-hand screw as a is rotated into b through the angle between the vectors.
In terms of the components of the original vectors,
a X b = (ya zb- yb za) i + (za xb- zb xa) j + (xa yb- xb ya) k
The components or direction numbers of the cross product may be evaluated using the determinant of a matrix .

 Cross Product (Vector Product) Dot Product (Scalar Product)

Dot Product: Also called the scalar product, defined by the formula a·b = |a| |b| cos (a,b). The angle (a,b) is the smallest angle measured between the common initial point of the vectors.
In terms of the components of the original vectors,
a·b = xa xb + ya yb + za zb
Given the components, and re-arranging the dot product formula, the angle between two vectors is :
(a,b) = arcos(a·b / |a| |b|)

Shortcuts

The Dot Product of perpendicular vectors is ZERO
The Cross Product of parallel vectors is ZERO
The Components of parallel vectors are PROPORTIONAL

Standard Angle Calculations

LINEAR ALGEBRA  contains the tables Vector Components and Direction Cosines. Values of (xa , ya , za) are expressed in terms of the trig functions of basic known angles SS, DD, and R1.

The worksheet Working Point Co-ordinates, listed under ANGLE CALCULATORS , calculates the values of the initial and terminal points of vectors, given the Main Pitch, Adjacent Pitch and Total Deck Angle.

Custom Calculations

Vector products may be applied to problems involving COMPOUND JOINERY : refer to the Compound Joinery Essay, and Alternate Convergent Joint Solution.

For the purpose of calculating angles on the surfaces of planes, the length of a vector is irrelevant, only the direction is important. For example, a vector lying on the x-axis may be replaced by any vector (x, 0, 0).

Vectors used to determine a cross product may possess any length and/or orientation, provided the vectors lie on the plane of interest. Furthermore, since there are actually two pairs of supplementary dihedral angles created by the intercept of two planes, cross products may be taken as a X b or b X a without affecting the resulting values of the dihedral angles.