Mathematics

Mathematics

Problem

Find the radius of the circular section in which the sphere ∣r∣=5|\mathbf{r}| = 5 is cut by the plane r⋅(i^+j^+k^)=33\mathbf{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 3\sqrt{3} .
Intuitive explanation

When a plane slices through a sphere, it creates a circular cross-section. The key is to find the perpendicular distance from the sphere's center to the plane, then use the Pythagorean theorem: the sphere's radius, the distance to the plane, and the circle's radius form a right triangle. Once you have that distance, you can solve for the radius of the circular section.

Intuitive explanation

When a plane slices through a sphere, it creates a circular cross-section. The key is to find the perpendicular distance from the sphere's center to the plane, then use the Pythagorean theorem: the sphere's radius, the distance to the plane, and the circle's radius form a right triangle. Once you have that distance, you can solve for the radius of the circular section.

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