Vectors: You are given two planes in parametric form....
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HallsofIvy
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You have two planes given by
\(\displaystyle \Pi_1:\begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix}= \begin{bmatrix}0 \\ 3 \\ -1 \end{bmatrix}+\lambda_1 \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}+ \lambda_2\begin{bmatrix}4 \\ -1 \\ 0 \end{bmatrix}\) and
\(\displaystyle \Pi_2:\begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix}= \begin{bmatrix}0 \\ 3 \\ 0 \end{bmatrix}+\mu_1 \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix}+ \mu_2\begin{bmatrix}4 \\ -1 \\ 1 \end{bmatrix}\)
\(\displaystyle \Pi_1:\begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix}= \begin{bmatrix}0 \\ 3 \\ -1 \end{bmatrix}+\lambda_1 \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}+ \lambda_2\begin{bmatrix}4 \\ -1 \\ 0 \end{bmatrix}\) and
\(\displaystyle \Pi_2:\begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix}= \begin{bmatrix}0 \\ 3 \\ 0 \end{bmatrix}+\mu_1 \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix}+ \mu_2\begin{bmatrix}4 \\ -1 \\ 1 \end{bmatrix}\)
pka
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To GetReal, I say get real yourself. Please learn to post readable questions. It is only thanks to Prof. Ivy that I have any idea what you are asking.
In the post \(\displaystyle \vec{p}+\lambda_1\vec{u}+\lambda_2\vec{v}\) is a plane that contains the point \(\displaystyle p\) and is spanned be \(\displaystyle \vec{u}~\&~\vec{v}\).
Sadly I tell you that notation is arcane at best. Almost all modern textbooks would give the same plane as;
\(\displaystyle {<x,y,z>}\cdot(\vec{u}\times\vec{v})=\vec{p}\cdot(\vec{u}\times\vec{v}) \)
In the post \(\displaystyle \vec{p}+\lambda_1\vec{u}+\lambda_2\vec{v}\) is a plane that contains the point \(\displaystyle p\) and is spanned be \(\displaystyle \vec{u}~\&~\vec{v}\).
Sadly I tell you that notation is arcane at best. Almost all modern textbooks would give the same plane as;
\(\displaystyle {<x,y,z>}\cdot(\vec{u}\times\vec{v})=\vec{p}\cdot(\vec{u}\times\vec{v}) \)