Vectors - Equations of Planes
Vectors - Equations of Planes
A plane \(p\) is parallel to that the lines
\({{l}_{1}}:r=\left( \begin{matrix} 3 \\0\\ -1 \end{matrix} \right)+\lambda \left( \begin{matrix} 4\\ -1\\1 \end{matrix} \right)\) and \({{l}_{2}}:r=\left( \begin{matrix}1\\2 \\-1 \end{matrix} \right)+\mu \left( \begin{matrix} 2 \\-5 \\-5 \end{matrix} \right)\).
Given that \({{l}_{2}}\) lies in the plane \(p\), find the equation of \(p\) in
- parametric form,
- scalar product form,
- Cartesian form.
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Answer:(a) \(\mathbf r=\begin{pmatrix}1\\2\\-1\end{pmatrix}+\lambda\begin{pmatrix}2\\-5\\-5\end{pmatrix}+\mu\begin{pmatrix}4\\-1\\1\end{pmatrix},\ \lambda,\mu\in\mathbb R\) (b) \(\mathbf r\cdot\begin{pmatrix}5\\11\\-9\end{pmatrix}=36\) (c) \(5x+11y-9z=36\)
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