taiwan-gsat 2024 Q12

taiwan-gsat · Other · ast__math-a 4 marks Vectors: Lines & Planes Coplanarity and Relative Position of Planes
In coordinate space, consider three planes $E_{1}: x + y + z = 7$, $E_{2}: x - y + z = 3$, $E_{3}: x - y - z = -5$. Let $L_{3}$ be the line of intersection of $E_{1}$ and $E_{2}$; $L_{1}$ be the line of intersection of $E_{2}$ and $E_{3}$; $L_{2}$ be the line of intersection of $E_{3}$ and $E_{1}$. It is known that the three lines $L_{1}, L_{2}, L_{3}$ have a common intersection point. Find the coordinates of this common intersection point $P$.
In coordinate space, consider three planes $E_{1}: x + y + z = 7$, $E_{2}: x - y + z = 3$, $E_{3}: x - y - z = -5$. Let $L_{3}$ be the line of intersection of $E_{1}$ and $E_{2}$; $L_{1}$ be the line of intersection of $E_{2}$ and $E_{3}$; $L_{2}$ be the line of intersection of $E_{3}$ and $E_{1}$.\\
It is known that the three lines $L_{1}, L_{2}, L_{3}$ have a common intersection point. Find the coordinates of this common intersection point $P$.