Let $\vec { F }$ Be a Constant Vector Field With ${ \vec { F } } = a \vec { i } + b \vec { j } + c \vec { k }$

Essay

Let $\vec { F }$ be a constant vector field with ${ \vec { F } } = a \vec { i } + b \vec { j } + c \vec { k }$ , where a, b, c are constants satisfying the condition $a ^ { 2 } + b ^ { 2 } + c ^ { 2 } = 1$ .Let S be a surface lying on the plane x + 4y - 5z = 10 oriented upward.
If the surface area of S is 10, what is the smallest possible value of $\int _ { S } \vec { F } \cdot \vec { d A }$ , and what are the corresponding values of a, b, c?