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}

$\frac{\mathrm{x}+1}{\mathrm{5}}=\frac{\mathrm{y}\u20132}{}={\lambda}_{2},\mathrm{z}=2$

\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 \ldots \\

3 \lambda_{1}-1=\lambda_{2}+1 \Rightarrow 3 \lambda_{1}-\lambda_{2}=2 \ldots

\end{array}

Solving (1) and (2),

$\begin{array}{l}\u201313{\lambda}_{2}=\u201310\\ {\lambda}_{2}=\frac{10}{13}\end{array}$\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.