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3D Coordinate Geometry

Append a z-axis to the 2-dimensional plane and conquer the realm of 3-dimensional space.

Intersection of Planes

         

State the relationship between the following two planes: \[\begin{align} -x + 3y -z &= 2 \\ x - 4y + z &= 2. \end{align}\]

What is the volume of the tetrahedron formed by the plane \[ 6x + 3y + 4z - 72 = 0 \] and the three coordinate planes?

If the following planes pass through the same line, what is the value of \( a+ b:\) \[ \begin{align} P_1 &: 3x + ay + 2z -2 =0 \\ P_2 &: 2x + by - 2z + a = 0 \\ P_3 &: 3x - y + 2z + b = 0? \end{align}\]

State the relationship between the following two planes: \[\begin{align} 2x - y + 2z &= -5 \\ x - 3y + 4z &= 3. \end{align}\]

State the relationship between the planes: \[\begin{align} 3x + y - 2z &= -4 \\ - 6x - 2y + 4z &= 1. \end{align}\]

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