Problemas Livro-2022 1

2. GEOMETRY: 4 problems 2/1. Let Σ₁ and Σ₂ be two parallel planes, the distance of which is equal to 8. We know that the intersection of a sphere S with Σ₁ is a circle of radius 5, the intersection of S with Σ₂ is a circle of radius 11. Compute the surface area of S. 2/2. Let O, A, B, C be points in the Euclidean space, not lying in a plane, a = \overrightarrow{OA}, b = \overrightarrow{OB}, c = \overrightarrow{OC}. Prove that the vector a × b + b × c + c × a is orthogonal to the plane spanned by the points A, B, and C. 2/3. Show that if a parallelogram is contained in a triangle, then its area is at most half of the area of the triangle. 2/4. Assume that a Cartesian coordinate system is fixed in the 3-dimensional Euclidean space. Compute the reflection of the point (1, 2, 3) in the plane defined by the equation 3x + 2y + z = 24.