Show that the plane 2x-y-3z=4 is parallel to the line

x= -2+2t

y= -1+4t

z= 4

and find the distance between them.

First, note that the line you give in parametric form can alternatively be writtenr= (-2, -1, 4) + t(2, 4, 0).

The vector (2, -1, -3) is normal or perpendicular to the plane 2x - y - 3z = 4.

The direction vector for the given line is (2, 4, 0).

To show that the line is parallel to the plane, show that the line's direction vector (2, 4, 0) is perpendicular to the plane's normal vector (2, -1, -3). That is, show that the (scalar) dot product of (2, 4, 0) and (2, -1, -3) is 0.

The dot product of (2, -1, -3) and (2, 4, 0) = 4 + (-4) + 0 = 0. Niiiiice.

Good luck!

-Andy