I'm having trouble with these types of questions. I have the following

rabbitz42z8 2021-11-13 Answered
I'm having trouble with these types of questions. I have the following vector \(\displaystyle{u}={\left({4},{7},-{9}\right)}\) and it wants me to find 2 vectors that are perpendicular to this one.
I know that \(\displaystyle{\left({4},{7},-{9}\right)}\cdot{\left({x},{y},{z}\right)}={0}\)

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Expert Answer

Opeance1951
Answered 2021-11-14 Author has 601 answers
Take vector \(\displaystyle{n}={\left({x},{y},{z}\right)}\). As you said by yourself, dot product should vanish. So
\(\displaystyle{\left({4},{7},-{9}\right)}\cdot{\left({x},{y},{z}\right)}={4}{x}+{7}{y}-{9}{z}={0}\)
As you might see, all points that lie on the plane \(\displaystyle{4}{x}+{7}{y}-{9}{z}={0}\) satisfy the condition of perpendicularity. If you want two linear independent vectors, just pick two different points. So, pick any two different triples \(\displaystyle{n}_{{1}}={\left({x}_{{1}},{y}_{{1}},\frac{{{4}{x}_{{1}}+{7}{y}_{{1}}}}{{9}}\right)}\) and \(\displaystyle{n}_{{2}}={\left({x}_{{2}},{y}_{{2}},\frac{{{4}{x}_{{2}}+{7}{y}_{{2}}}}{{9}}\right)}\) where \(\displaystyle{\left({x}_{{1}},{y}_{{1}}\right)}\ne{\left({x}_{{2}},{y}_{{2}}\right)}\), and you're done.
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