Show that the lines and do not intersect each other.
Show that the lines and do not intersect each other.

Show that the lines and do not intersect each other.
Given: The equations of the two lines are and
To Prove: the lines do not intersect each other.
Formula Used: Equation of a line is
Vector form:
Cartesian form:
where is a point on the line and is the direction ratios of the line.
Proof:
Let

$\frac{x–1}{3}=\frac{y+1}{3}=z={\lambda }_{1}$
\frac{x-1}{3}=\frac{y+1}{3}=z=\lambda_{1}
$\frac{\mathrm{x}+1}{\mathrm{5}}=\frac{\mathrm{y}–2}{}={\lambda }_{2},\mathrm{z}=2$
\frac{\mathrm{x}+1}{\mathrm{5}}=\frac{\mathrm{y}-2}{\mathfrak{}}=\lambda_{2}, \mathrm{z}=2

So a point on the first line is
A point on the second line is
If they intersect they should have a common point.

$\begin{array}{l}2{\lambda }_{1}+1=5{\lambda }_{2}–1⇒2{\lambda }_{1}–5{\lambda }_{2}=–2\dots \\ 3{\lambda }_{1}–1={\lambda }_{2}+1⇒3{\lambda }_{1}–{\lambda }_{2}=2\dots \end{array}$
\begin{array}{l}
2 \lambda_{1}+1=5 \lambda_{2}-1 \Rightarrow 2 \lambda_{1}-5 \lambda_{2}=-2 \ldots \\
3 \lambda_{1}-1=\lambda_{2}+1 \Rightarrow 3 \lambda_{1}-\lambda_{2}=2 \ldots
\end{array}

Solving (1) and (2),

$\begin{array}{l}–13{\lambda }_{2}=–10\\ {\lambda }_{2}=\frac{10}{13}\end{array}$
\begin{array}{l}
-13 \lambda_{2}=-10 \\
\lambda_{2}=\frac{10}{13}
\end{array}

Therefore,
Substituting for the z coordinate, we get
and
So, the lines do not intersect.