In regular tetrahedral oabc with edge length 1, if point P satisfies vector OP = x vector OA + y vector ob + Z vector OC, and X + y + Z = 1, then what is the minimum value of modulus of vector OP? It is related to the combination of number and shape
Let the center of gravity of triangle ABC be m
OA=OM+MA
OB=OM+MB
OC=OM+MC
Vector OP = x vector OA + y vector ob + Z vector OC
=(x + y + Z) * vector om + X vector Ma + y vector MB + Z vector MC
=Vector om + X vector Ma + y vector MB + Z vector MC
Then x vector Ma + y vector MB + Z vector MC = vector MP
If M is in plane ABC, then p is in plane ABC
Then the minimum value of the module of the vector OP is the distance from O to the plane ABC
According to Pythagorean theorem, the minimum value of the module of OP is (radical 3) / 6
When x = 1 / 3, y = 1 / 3, z = 1 / 3, take the minimum