Geometry

3D Coordinate Geometry

3D Coordinate Geometry: Level 3 Challenges

         

What is the shortest distance of the plane 4x3y+12z=78 4x - 3y + 12 z= 78 from the origin in R3 \mathbb{R}^{3}?

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

What is the volume of this cuboid?

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

Line \ell goes through the origin OO and intersects TT perpendicularly in point PP.

Determine the distance OPOP.

The graph x2+y2=z23x^2 + y^2 = \dfrac{z^2}{3} intersects with the graph z=ky+cz = ky + c, where kk and cc are real numbers, such that the intersected points form a parabola with the point (0,12,32)\left (0 , \dfrac{-1}{2}, \dfrac{\sqrt{3}}{2}\right) as its vertex.

What is the value of k2k^2?

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

Submit answer as the sum of the coordinates.

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