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

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