Write the plane through \((1,2,3)\) with normal \((2,-1,1)\).

Simplify \(2(x-1)-(y-2)+(z-3)=0\).

Does \((1,2,3)\) lie on \(2x-y+z=3\)?

Does \((0,0,0)\) lie on \(2x-y+z=3\)?

Find a normal vector to \(3x+4y-5z=7\).

Find a plane through the origin with normal \((1,1,1)\).

Find the plane through \(A=(1,0,0)\), \(B=(0,1,0)\), and \(C=(0,0,1)\).

Find the intersection point of line \(\mathbf r=(0,0,0)+t(1,2,3)\) with plane \(x+y+z=6\).

Find the distance from \((1,1,1)\) to plane \(2x-y+2z=4\).

Are planes \(x+2y+3z=4\) and \(2x+4y+6z=9\) parallel?

Find the angle between planes with normals \((1,0,1)\) and \((0,1,1)\).

Find a parametric equation for plane \(x+y+z=1\).

Find the normal to the plane spanned by \((1,2,0)\) and \((0,1,3)\).

Find the plane through \((2,0,1)\) parallel to \(x-y+z=5\).

A line is given by \(\mathbf r=(1,-1,0)+t(1,2,1)\). Find where it meets the plane \(2x+3y+z=5\).

The line \(\mathbf r=(2,0,1)+t(1,-1,0)\) has direction parallel to the plane \(x+y+z=2\). Decide whether the whole line lies in the plane or is parallel but separate.

Find the line of intersection of planes \(x+y=1\) and \(z=2\), then verify that its direction is perpendicular to both plane normals.

Find the plane through \(A=(1,0,2)\), \(B=(3,1,0)\), and \(C=(0,2,1)\), then test whether \(P=(2,1,1)\) lies on it.

Find the foot of the perpendicular from \(P=(1,2,3)\) to the plane \(2x-2y+z=6\), and give the distance from \(P\) to the plane.

A slab is bounded by the parallel planes \(2x-y+2z=4\) and \(2x-y+2z=10\). Find its thickness and a point on the second plane reached by moving normally from \((2,0,0)\) on the first plane.