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

$\begin{array}{l}2{\lambda }_1+1=5{\lambda }_2–1\Rightarrow 2{\lambda }_1–5{\lambda }_2=–2\dots \\ 3{\lambda }_1–1={\lambda }_2+1\Rightarrow 3{\lambda }_1–{\lambda }_2=2\dots \end{array}$
\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.