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3D Coordinate Geometry

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

Parallel Planes

         

Which of the following is/are true regarding three planes \(\alpha,\beta,\) and \(\gamma\) in the coordinate space?

I. If \(\alpha:~a_1x+b_1y+c_1z+d_1=0\) and \(\beta:~a_2x+b_2y+c_2z+d_2=0\) are parallel, then \(a_1=a_2,~b_1=b_2,\) and \(c_1=c_2.\)

II. If \(\alpha\parallel\beta\) and \(\beta\parallel\gamma,\) then \(\alpha\parallel\gamma.\)

III. If \(\alpha\bot\beta\) and \(\beta\bot\gamma,\) then \(\alpha\parallel\gamma.\)

If the two planes \(18x+15y-6z+1=0\) and \(ax+by+2z+1=0\) are parallel, what is the value of \(ab?\)

What is the distance between the two planes \(\alpha:~2x-y+3z+4=0\) and \(\beta:~2x-y+3z-24=0?\)

What is the equation of the plane that is parallel to \(3x+2y+z-1=0\) and passes through the point \(P=(-5,1,-5)?\)

Which of the following planes is parallel to \[x-2y+3z-4=0?\]

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