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Mathematik-Online problems:

Problem 15: Intersections and Distances of Lines and Planes

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
The points $ A=(-1, 1, 1)$ , $ B=(2, 3, 0)$ and $ C=(0, 1, -2)$ define a plane $ E_1$ in $ \mathbb{R}^3$ .
a)
Find the Hessian normal form of $ E_1$ .
b)
Find a parameter representation of the line $ g_1$ through the point $ P=(1, 0, -2)$ , which is orthogonal to $ E_1$ .
c)
What's the distance between $ P$ and $ E_1$ ?
d)
What's the angle between $ E_1$ and the plane $ E_2: x_1+x_2+x_3=1$ ?
e)
The line

$\displaystyle g_2: x=\left(\begin{array}{r} -5\\ -8\\ 11\end{array}\right)+\lam... ...ft(\begin{array}{r} 1\\ 1\\ -1\end{array}\right), \qquad \lambda\in\mathbb{R}, $


intersects with $ E_1$ in a point $ S$ . Find $ S$ and the image line $ g'_2$ of $ g_2$ by reflection about $ E_1$ .
f)
What's the distance between $ S$ and $ g_1$ ?

(Authors: Kimmerle/Roggenkamp/Höfert)

Solution:

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  automatisch erstellt am 9.  5. 2008