4. Line and Plane

Line and Plane in Space

Ex 1.

Check if the line given parametrically

\[\ell: \; x=1+2t,\; y= -1+t,\; z=3- t\]

intersects the plane \(\pi: 2x-y+z-4=0\). If so, provide the intersection point.

Ex 2.

\(\star\) Calculate the distance of point \(G(2,-1,0)\) from the line passing through points \(H(0,0,0)\) and \(I(1,1,1)\).

Ex 3.

\(\star\) Consider the system of a line and a plane dependent on parameter \(\lambda\):

\[\ell(\lambda): x=\lambda+t,\; y=1+2t,\; z=2-t\]

and

\[\pi: x - (\lambda-1) y + z - 3 = 0\]

Determine the values of \(\lambda\) for which the line is parallel to the plane, contained in the plane, or intersects it at a single point.