Math Problem Statement

The equations of two lines are: \[ r = \begin{pmatrix} 3 \\ 5 - c \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} \text{ and } r = \begin{pmatrix} 5 \\ 2 \\ 2c - 3 \end{pmatrix} + t \begin{pmatrix} c - 1 \\ 4 \\ -1 \end{pmatrix}, \] where \( c \in \mathbb{R} \). Given that the lines intersect, determine the possible values of \( c \).

Solution

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Equations of Lines
Intersection of Lines

Formulas

Parametric equation of a line
Solving simultaneous equations

Theorems

Conditions for the intersection of lines in 3D

Suitable Grade Level

Grades 11-12

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