3. Planes

Planes

Ex 1.

Provide the general and normal equation of the plane passing through point \(A(1,0,2)\) and with the normal vector \(\mathbf{n}=[2,-1,1]\).

Ex 2.

Find the equation of the plane passing through points \(A(1,0,0)\), \(B(0,1,0)\), and \(C(0,0,1)\).

Ex 3.

Determine the angle between the planes: \(\pi_1: x+2y-2z+1=0\) and \(\pi_2: 2x-y+z-3=0\).

Ex 4.

For the plane \(\pi: x-2y+2z-4=0\), calculate the distance of point \(P(3,0,1)\) from this plane.

Ex 5.

Find a vector perpendicular to the plane \(x + y + z = 1\).

Ex 6.

A plane passes through point \(A(1, 2, 3)\) and is parallel to the plane \(2x + 3y + 4z = 5\). Find the equation of this plane.

Ex 7.

\(\star\) Find the equation of the plane passing through point \(D(1,1,1)\) and containing the line passing through points \(E(0,0,0)\) and \(F(1,2,3)\).