Geometry

# 3D Coordinate Geometry - 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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