3D Coordinate Geometry - Problem Solving
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Assume a particle is at the origin of a 3D plane. If it moves 5 units in the direction of the \(x\)-axis, then 12 units in the direction of the \(y\)-axis, and then 84 units in the direction of the \(z\)-axis, what is its displacement?
By the Pythagoras theorem, the displacement is \(\sqrt{5^2 + 12^2+ 84^2} = 85. \ _\square\)
In 3-dimensional space, what is the number of planes that are equidistant from four non-coplanar points?
An ellipsoid with center \((0,0,0)\) and semi-axes of lengths 10, 15, and 30 along the \(x, y,\) and \(z\) directions, respectively, is cut by a plane whose equation is \( x + 3y + 2z = 40 \).
The intersection of the ellipsoid and the plane is an ellipse.
Find its area.
This problem is original.