Plane one

Plane Two

12 Three Dimensional Geometry
Skew Lines and Shortest Distance
between them

Use mouse pointer to have animations
Drag H and J to move vectors
You may drag D also.Use check boxes for parallel planes

Start Animation

\overrightarrow{b_1}×\overrightarrow{b_2}

Stop

\overrightarrow{r}=\overrightarrow{a_1}+λ\overrightarrow{b_1}

\overrightarrow{r}=\overrightarrow{a_2}+μ\overrightarrow{b_2}

\overrightarrow{b_1}

\overrightarrow{b_2}

Shortest Distance between two skew lines (SD)
is the projection of the vector AB on SD line 
(yellow dotted line)

\overrightarrow{a_2}-\overrightarrow{a_1}

\overrightarrow{AB}=\overrightarrow{a_2}-\overrightarrow{a_1}

a: x + (2 * y) + z = 7
S = PointIn(a)
f = OrthogonalLine(S, a)
A = PointIn(a)
g = Line(S, A)
D = Point(f)
e = Plane(D, a)
B = PointIn(e)
h = Line(B, D)
i = Line(A, B)
u = Vector(A, B)
j = Segment(S, D)
k = false
b = false
C = Point(g)
E = Point(j)
F = Point(h)
G = Point(j)
α = Angle(E, S, C)
β = Angle(F, D, G)
H = Point(j)
I = Point(g)
v = Vector(S, I)
H' = Translate(H, v)
H'_1 = Translate(H, v)
d = Vector(H, H'_1)
J = Point(j)
K = Point(h)
w = Vector(D, K)
J' = Translate(J, w)
c = Vector(J, J')
l = Vector(H, D)
L = Midpoint(H, D)
M = Point(g)
N = Point(h)
m = Line(D, S)
O = Midpoint(A, B)