Vectors - Equations of Planes
Vectors - Equations of Planes
A plane \(p\) passes through point \(A(1,-2,3)\). Given that the line
\(l:r=\left( \begin{matrix} 1 \\ 1 \\ -4 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 2 \\ 1 \\ 1 \\ \end{matrix} \right)\) lies in the plane \(p\). Find the equation of \(p\) in
- parametric form,
- scalar product form,
- Cartesian form.
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Answer:(a) \(\mathbf r=\begin{pmatrix}1\\-2\\3\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\1\end{pmatrix}+\mu\begin{pmatrix}0\\3\\-7\end{pmatrix},\ \lambda,\mu\in\mathbb R\) (b) \(\mathbf r\cdot\begin{pmatrix}-5\\7\\3\end{pmatrix}=-10\) (c) \(-5x+7y+3z=-10\)
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