(Solved): 4. Consider the points \( A(1,1,1) \) and \( B(0,-2,1) \) and \( O(0,0,0) \). (a) Find a nonzero v ...

4. Consider the points \( A(1,1,1) \) and \( B(0,-2,1) \) and \( O(0,0,0) \). (a) Find a nonzero vector orthogonal to the plane containing the three points. (b) Write down a scalar equation for the plane containing the three points. (c) Let \( \vec{a}=\overrightarrow{O A} \) and \( \vec{b}=\overrightarrow{O B} \). Show that the point \( P \) corresponding to the vector \( \vec{p} \), where \[ \vec{p}=s \vec{a}+t \vec{b}, \] is a point on the plane for every \( s \) and \( t \). (d) Conclude that the plane is given by \( \operatorname{span}\{\vec{a}, \vec{b}\} \).