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eguruchela

List of formulas related to Analytic-geometry (Line in 3D)


Line forms

Point direction form:

xx1a=yy1b=zz1c

Two point form:

xx1x2x1=yy1y2y1=zz1z2z1

Parametric form:

x=x1+tcosαy=y1+tcosβz=z1+tcosγ

Distance between two lines in 3 dimensions

The distance from P2(x2,y2,z2)
to the line through P1(x1,y1,z1)
in the direction (a,b,c) is
d=[c(y2y1)b(z2z1)]2+[a(z2z1)c(x2x1)]2+[b(x2x1)a(y2y1)]2a2+b2+c2

The distance between two lines.

First one: through P1(x1,y1,z1)
in direction (a1,b1,c1),
Second one: through P2(x2,y2,z2)
in direction (a2,b2,c2) is:
d=|x2x1y2y1z2z1a1b1c1a2b2c2||b1c1b2c2|2+|c1a1c2a2|2+|a1b1a2b2|2

The two lines intersect if:

|x2x1y2y1z2z1a1b1c1a2b2c2|=0