Find coordinates for two points that belong to the plane 2x+3y+5z=15. Show that the vector [2,3,5] is perpendicular to the segment that joins your two points. Explain why [2,3,5] is perpendicular to the plane.

Answer 1

how about the two points
(0,0,3) and (3,3,0) ?

a line containing those two point has direction
[3,3,-3] or [1,1,-1] if reduced

let's take the dot product of [1,1,-1] and [2,3,5]
= 2 + 3 - 5
= 0
so if the dot product is 0, then the vectors are perpendicular.

By now you should have learned that for the plane
ax + by + cz = d
[a,b,c] is a normal to the plane.

Answer 2

what does (a,b,c) is normal to the plane mean? specifically, what do u mean by "normal"?