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# 3D Coordinate Geometry

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

# 3D Coordinate Geometry: Level 3 Challenges

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

###### Image credit: Wikimedia Commons: Sakurambo

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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