Prove that the lines and intersect each other and find the point of their intersection.

Answer

Given: The equations of the two lines are and

To Prove: The two lines intersect and to find their point of intersection.

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

\begin{array}{l}

\frac{x-4}{1}=\frac{y+3}{1}=\frac{z+1}{7}=\lambda_{1} \\

\frac{x-1}{\sqrt{1}}=\frac{y+1}{-3}=\frac{z+10}{8}=\lambda_{2}

\end{array}

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}

\lambda_{1}+4=2 \lambda_{2}+1 \Rightarrow \lambda_{1}-2 \lambda_{2}=-3 \ldots \\

4 \lambda_{1}-3=-3 \lambda_{2}-1 \Rightarrow 4 \lambda_{1}+3 \lambda_{2}=2 \ldots

\end{array}

Solving (1) and (2),

$\begin{array}{l}11{\lambda}_{2}=14\\ {\lambda}_{2}=\frac{14}{11}\end{array}$\begin{array}{l}

11 \lambda_{2}=14 \\

\lambda_{2}=\frac{14}{11}

\end{array}

Therefore,

Substituting for the z coordinate, we get

and

So, the lines do not intersect.