Lines in Three Dimensions

Line forms

Point direction form:

\( \frac{x-x_1}{a} = \frac{y – y_1}{b} = \frac{z-z_1}{c} \)

Two point form:

\( \frac{x-x_1}{x_2-x_1} = \frac{y – y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1} \)

Parametric form:

\(
\begin{aligned}
x &= x_1 +t\,\cos \alpha \\
y &= y_1 +t\,\cos \beta \\
z &= z_1 +t\,\cos \gamma
\end{aligned}
\)

Distance between two lines in 3 dimensions

The distance from \(P_2(x_2,y_2,z_2)\) to the line through \(P_1(x_1,y_1,z_1)\) in the direction (a,b,c) is

\(
d = \sqrt{ \frac{\left[c(y_2-y_1)-b(z_2-z_1)\right]^2 +
\left[a(z_2-z_1)-c(x_2-x_1)\right]^2 +
\left[b(x_2-x_1)-a(y_2-y_1)\right]^2} {a^2 + b^2 + c^2 } }
\)

The distance between two lines. First one through \(P_1(x_1,y_1,z_1)\) in direction \((a_1,b_1,c_1)\). Second one: through \(P_2(x_2,y_2,z_2)\) in direction \((a_2,b_2,c_2)\) is:

\(
d = \frac{ \begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} }
{ \sqrt{\begin{vmatrix} b_1 & c_1 \\ b_2 & c_2 \end{vmatrix}^2 +
\begin{vmatrix} c_1 & a_1 \\ c_2 & a_2 \end{vmatrix}^2 +
\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}^2 }}
\)

The two lines intersect if:

\( \begin{vmatrix} x_2-x_1 & y_2 – y_1 & z_2 – z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = 0 \)

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