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

What is the shortest distance of the plane \( 4x - 3y + 12 z= 78\) from the origin in \( \mathbb{R}^{3}\)?

In the \(xyz\) coordinates, the centroids of the 3 faces of a cuboid are located at points \((2 , 2 , 2), (7 , 5 , 2)\), and \((7 , 2 , 4)\), where each side is parallel to one of the axes.

What is the volume of this cuboid?

Let \(T\) be the triangle with vertices \((14, 0, 0)\), \((0, 21, 0)\), and \((0, 0, 42)\).

Line \(\ell\) goes through the origin \(O\) and intersects \(T\) perpendicularly in point \(P\).

Determine the distance \(OP\).

The graph \(x^2 + y^2 = \dfrac{z^2}{3}\) intersects with the graph \(z = ky + c\), where \(k\) and \(c\) are real numbers, such that the intersected points form a parabola with the point \(\left (0 , \dfrac{-1}{2}, \dfrac{\sqrt{3}}{2}\right)\) as its vertex.

What is the value of \(k^2\)?

Find the coordinates of the point where the line through \((5, 1, 6)\) and \((3, 4, 1)\) crosses the \(YZ\)-plane

Submit answer as the sum of the coordinates.

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