find the distance between plane 1 : 2x-3y+z=5 and plane 2: x+2y+3z=2

Question

This is a question on vector in euclidean 3 space

Answer 1

If two planes are not parallel then the shortest distance between them is zero, because they intercept one another along a line.

The equations of the planes relate the variables so we can write one variable in terms of the other two. If we double the equation for plane 2 we get 2x+4y+6z=4, then substitute for 2x from the other equation we get 5+3y-z+4y+6z=4; 7y+5z+1=0. This is the equation of a line where the x coordinate is the same for each plane. This would appear to be the line of intersection, so the shortest distance between the planes is 0.