Vectors - Equations of Planes

Vectors - Equations of Planes

A plane \(p\) passes through point \(A\left( 3,2,-5 \right)\). Given that the line \(l:r=\left( \begin{matrix} 1 \\ 0 \\ -2 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 5 \\ 2 \\ -2 \\ \end{matrix} \right)\) lies in the plane \(p\). Find the equation of \(p\) in

  1. parametric form,
  2. scalar product form,
  3. Cartesian form.

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Answer:(a) \(\mathbf r=\begin{pmatrix}1\\0\\-2\end{pmatrix}+\lambda\begin{pmatrix}5\\2\\-2\end{pmatrix}+\mu\begin{pmatrix}-2\\-2\\3\end{pmatrix},\ \lambda,\mu\in\mathbb R\) (b) \(\mathbf r\cdot\begin{pmatrix}2\\-11\\-6\end{pmatrix}=14\) (c) \(2x-11y-6z=14\)

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