Find an equation for the plane containing the two (parallel)

eozoischgc 2022-01-07 Answered
Find an equation for the plane containing the two (parallel) lines
\(\displaystyle{v}_{{1}}={\left({0},{1},−{2}\right)}+{t}{\left({2},{3},−{1}\right)}\)
\(\displaystyle{v}_{{2}}={\left({2},−{1},{0}\right)}+{t}{\left({2},{3},−{1}\right)}\)

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Expert Answer

Joseph Lewis
Answered 2022-01-08 Author has 4556 answers

Given:
\(\displaystyle{v}_{{1}}={\left({0},{1},−{2}\right)}+{t}{\left({2},{3},−{1}\right)}\)
\(\displaystyle{v}_{{2}}={\left({2},−{1},{0}\right)}+{t}{\left({2},{3},−{1}\right)}\)
Let \(\displaystyle{A}={\left({0},{1},-{2}\right)}\), \(\displaystyle{B}={\left({2},-{1},{0}\right)}\)
Thus, the plane containing these two lines will contain A and B. Hence, the vector \(\displaystyle{A}{B}={\left({2},-{2},{2}\right)}\)
Normal to the plave shall be the cross product of AB and the direction ratio of the lines.
Thus, \(\displaystyle\vec{{{n}}}=\vec{{{A}{B}}}\times{d}{i}{r}{e}{c}{t}{i}{o}{n}\ {r}{a}{t}{i}{o}\ {o}{f}\ {t}{h}{e}\ {l}in{e}\)
\(\displaystyle\vec{{{n}}}={\left({2},-{2},{2}\right)}\times{\left({2},{3},-{1}\right)}\)
\[\vec{n}=\begin{bmatrix}i & j & k \\2 & -2 & 2 \\2 & 3 & -1 \end{bmatrix}=-4i+6j+10k\]
Thus, the equation of the plane can be written as
\(\displaystyle{\left({x}-{o}\right)}{i}+{\left({y}-{1}\right)}{j}+{\left({z}+{2}\right)}\times\vec{{{n}}}={0}\)
\(\displaystyle-{4}{\left({x}-{0}\right)}+{6}{\left({y}-{1}\right)}+{10}{\left({z}+{2}\right)}={0}\)
\(\displaystyle-{4}{x}+{6}{y}+{10}{z}+{14}={0}\)

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