Express the equation of the lines

\(\displaystyle{o}{v}{e}{r}\rightarrow{\left\lbrace{r}\right\rbrace}\cdot{\left({w}{i}{d}{e}\hat{{{i}}}-{w}{i}{d}{e}\hat{{{j}}}+{7}={0},{o}{v}{e}{r}\rightarrow{\left\lbrace{r}\right\rbrace}\cdot{\left({w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{j}}}\right)}-{5}={0}\right.}\)

in parametric form

\(\displaystyle{L}{e}{t}{w}{i}{d}{e}\hat{{{r}}}={x}{w}{i}{d}{e}\hat{{{i}}}+{y}{w}{i}{d}{e}\hat{{{j}}}+{z}{w}{i}{d}{e}\hat{{{k}}}\)

\(\displaystyle{w}{i}{d}{e}\hat{{{r}}}\cdot{\left({w}{i}{d}{e}\hat{{{i}}}-{w}{i}{d}{e}\hat{{{j}}}\right)}+{7}={0}\)

\(\displaystyle{\left({x}{w}{i}{d}{e}\hat{{{i}}}+{y}{w}{i}{d}{e}\hat{{{j}}}+{z}{w}{i}{d}{e}\hat{{{k}}}\right)}{\left({w}{i}{d}{e}\hat{{{i}}}{w}{i}{d}{e}\hat{{{j}}}\right)}+{7}={0}\)

\(\displaystyle{L}\in{e}{L}_{{1}}:-{x}-{y}+{7}={0}\)

\(\displaystyle{A}{n}{d}{w}{i}{d}{e}\hat{{{r}}}\cdot{\left({w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{j}}}\right)}-{5}={0}\)

\(\displaystyle{\left({x}{w}{i}{d}{e}\hat{{{i}}}+{y}{w}{i}{d}{e}\hat{{{j}}}+{z}{w}{i}{d}{e}\hat{{{k}}}\right)}{\left({w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{a}{b}}}\right)}-{5}={0}\)

\(\displaystyle{L}\in{e}{L}_{{2}}{x}+{3}{y}-{5}={0}\)

Line \(\displaystyle{L}_{{1}}\) is in direction of \(\displaystyle{w}{i}{d}{e}\hat{{{i}}}-{w}{i}{d}{e}\hat{{{j}}}--{\left({v}\right)}\)

And \(\displaystyle{L}_{{1}}\) also passes through (-3,4)(A)

Vector equation of \(\displaystyle{L}_{{1}}\Rightarrow{X}={A}+{t}{V}\)

\(\displaystyle{\left[{x},{y}\right]}={\left[-{3},{4}\right]}+{t}{\left[{1},{1}\right]}\)

\(\displaystyle{\left[{x},{y}\right]}={\left[-{3},{4}\right]}+{t}{\left[{t},{t}\right]}\)

\(\displaystyle{\left[{x},{y}\right]}={\left[-{3},+{t},{4}+{t}\right]}\)

in parametric form

\(\displaystyle{<}{x},{y}\ge{<}-{3}+{t},{4}+{t}{>}\)

Similarly,line \(\displaystyle{L}_{{2}}\) also passes through (2,1)(A)

and direction vector [-3,1](v)

\(\displaystyle{L}_{{2}}={x}={a}+{t}{V}\)

\(\displaystyle{\left[{x},{y}\right]}={\left[{2},{1}\right]}+{t}_{{1}}{\left[-{3},{1}\right]}\)

\(\displaystyle{\left[{x},{y}\right]}={\left[{2}-{3}{t}_{{1}},{1}+{t}_{{1}}\right]}\)

\(\displaystyle{L}_{{2}}:\prec{x},{y}\ge{<}{2}-{3}{t}_{{1}},{1}+{t}_{{1}}{>}\)

Point of intersection of \(\displaystyle{L}_{{1}}{\quad\text{and}\quad}{L}_{{2}}\) are:-

x=-4

So,Position vecor of their point of intersection

\(\displaystyle=-{4}{w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{j}}}\)

\(\displaystyle{o}{v}{e}{r}\rightarrow{\left\lbrace{r}\right\rbrace}\cdot{\left({w}{i}{d}{e}\hat{{{i}}}-{w}{i}{d}{e}\hat{{{j}}}+{7}={0},{o}{v}{e}{r}\rightarrow{\left\lbrace{r}\right\rbrace}\cdot{\left({w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{j}}}\right)}-{5}={0}\right.}\)

in parametric form

\(\displaystyle{L}{e}{t}{w}{i}{d}{e}\hat{{{r}}}={x}{w}{i}{d}{e}\hat{{{i}}}+{y}{w}{i}{d}{e}\hat{{{j}}}+{z}{w}{i}{d}{e}\hat{{{k}}}\)

\(\displaystyle{w}{i}{d}{e}\hat{{{r}}}\cdot{\left({w}{i}{d}{e}\hat{{{i}}}-{w}{i}{d}{e}\hat{{{j}}}\right)}+{7}={0}\)

\(\displaystyle{\left({x}{w}{i}{d}{e}\hat{{{i}}}+{y}{w}{i}{d}{e}\hat{{{j}}}+{z}{w}{i}{d}{e}\hat{{{k}}}\right)}{\left({w}{i}{d}{e}\hat{{{i}}}{w}{i}{d}{e}\hat{{{j}}}\right)}+{7}={0}\)

\(\displaystyle{L}\in{e}{L}_{{1}}:-{x}-{y}+{7}={0}\)

\(\displaystyle{A}{n}{d}{w}{i}{d}{e}\hat{{{r}}}\cdot{\left({w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{j}}}\right)}-{5}={0}\)

\(\displaystyle{\left({x}{w}{i}{d}{e}\hat{{{i}}}+{y}{w}{i}{d}{e}\hat{{{j}}}+{z}{w}{i}{d}{e}\hat{{{k}}}\right)}{\left({w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{a}{b}}}\right)}-{5}={0}\)

\(\displaystyle{L}\in{e}{L}_{{2}}{x}+{3}{y}-{5}={0}\)

Line \(\displaystyle{L}_{{1}}\) is in direction of \(\displaystyle{w}{i}{d}{e}\hat{{{i}}}-{w}{i}{d}{e}\hat{{{j}}}--{\left({v}\right)}\)

And \(\displaystyle{L}_{{1}}\) also passes through (-3,4)(A)

Vector equation of \(\displaystyle{L}_{{1}}\Rightarrow{X}={A}+{t}{V}\)

\(\displaystyle{\left[{x},{y}\right]}={\left[-{3},{4}\right]}+{t}{\left[{1},{1}\right]}\)

\(\displaystyle{\left[{x},{y}\right]}={\left[-{3},{4}\right]}+{t}{\left[{t},{t}\right]}\)

\(\displaystyle{\left[{x},{y}\right]}={\left[-{3},+{t},{4}+{t}\right]}\)

in parametric form

\(\displaystyle{<}{x},{y}\ge{<}-{3}+{t},{4}+{t}{>}\)

Similarly,line \(\displaystyle{L}_{{2}}\) also passes through (2,1)(A)

and direction vector [-3,1](v)

\(\displaystyle{L}_{{2}}={x}={a}+{t}{V}\)

\(\displaystyle{\left[{x},{y}\right]}={\left[{2},{1}\right]}+{t}_{{1}}{\left[-{3},{1}\right]}\)

\(\displaystyle{\left[{x},{y}\right]}={\left[{2}-{3}{t}_{{1}},{1}+{t}_{{1}}\right]}\)

\(\displaystyle{L}_{{2}}:\prec{x},{y}\ge{<}{2}-{3}{t}_{{1}},{1}+{t}_{{1}}{>}\)

Point of intersection of \(\displaystyle{L}_{{1}}{\quad\text{and}\quad}{L}_{{2}}\) are:-

x=-4

So,Position vecor of their point of intersection

\(\displaystyle=-{4}{w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{j}}}\)