https://file.aiquickdraw.com/mathbot/file/1727381523314vnvuccnj.jpg

The task involves finding the intersection points between a line \( g \) and a plane \( E \). The parametric equation of the line \( g \) is given as:

\[ \mathbf{x} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix} \]

And the equation of the plane \( E \) is provided in a parametric form:

\[ \mathbf{x} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} + r \begin{pmatrix} 4 \\ -1 \\ -1 \end{pmatrix} + s \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix} \]

To find the intersection points, we need to set the parametric equation of the line equal to the parametric equation of the plane and solve for \( t \), \( r \), and \( s \).

Substituting the equation of the line into the equation of the plane, we get:

\[ \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} + r \begin{pmatrix} 4 \\ -1 \\ -1 \end{pmatrix} + s \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix} \]

We will solve this system to find the values of \( t \), \( r \), and \( s \). Let's start the calculations.

Bestimmen Sie die gemeinsamen Punkte der Geraden g: \( \mathbf{x} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix} \) und der Ebene E. Bestimmen Sie zunächst eine Koordinatengleichung von E.

Learn how to determine the intersection points between a line and a plane using parametric equations. This problem covers key concepts in analytical geometry, suitable for students in grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation