Express the equations of the lines r · (i −j) + 7 = 0,r · (i + 3j) − 5 = 0 in parametric forms and hence find the position vector of their point of intersection.

Question
Forms of linear equations
asked 2020-11-02
Express the equations of the lines
r · (i −j) + 7 = 0,r · (i + 3j) − 5 = 0 in parametric forms and hence find the position
vector of their point of intersection.

Answers (1)

2020-11-03
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
\(\displaystyle=-{4}{w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{j}}}\)
0

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